Chinnaraji Annamalai

Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management

SCHOLARLY PAPERS

70

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Ideas:
“  I am currently working on computing techniques and computational methods based on the combinatorial geometric series, binomial and multinational theorems and also factorials with integers for the applications in science, engineering and management.  ”

Scholarly Papers (70)

1.

Annamalai's Binomial Identity and Theorem

Number of pages: 7 Posted: 05 May 2022 Last Revised: 23 Oct 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 561 (72,827)

Abstract:

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extended geometric series, Annamalai’s binomial theorem, binomial coefficient

2.

A Model of Iterative Computations for Recursive Summability

IJSART - Volume 5 Issue 5 –MAY 2019
Number of pages: 2 Posted: 06 Jun 2019
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 204 (217,526)

Abstract:

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recursive algorithm, real-time system, iterative computation

3.

Computation of Series of Series Using Annamalai’s Computing Model

Number of pages: 3 Posted: 27 Jun 2019
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 193 (228,511)

Abstract:

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numerical computation, summability, series of finite series

4.

Novel Binomial Series and its Summations

Number of pages: 3 Posted: 13 May 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 124 (327,233)

Abstract:

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Binomial coefficient, binomial series, optimized combination

5.

Intuitionistic Fuzzy Sets: New Approach and Applications

International Journal of Research in Computer and Communication Technology, 2014
Number of pages: 4 Posted: 24 May 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 111 (353,872)

Abstract:

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Fuzzy set; Intuitionistic fuzzy set; Domain of discourse

6.

Sum of Geometric Series with Negative Exponents

Number of pages: 1 Posted: 13 May 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 109 (358,311)

Abstract:

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geometric series, summation, successive terms, negative exponents

7.

My New Idea for Optimized Combinatorial Techniques

HAL (2022)
Number of pages: 4 Posted: 20 Jun 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 108 (360,658)

Abstract:

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algorithm, combinatorics, computation

8.

Comparison between Optimized and Traditional Combinations of Combinatorics

Number of pages: 2 Posted: 13 May 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 93 (397,516)

Abstract:

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optimized combination, binomial coefficient, combinatorics

9.

Application of Factorial and Binomial identities inCybersecurity

Number of pages: 2 Posted: 03 Jun 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 91 (402,989)

Abstract:

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discrete mathematics, theoretical computer science, binomial identity, cybersecurity, factorial, computing, networks

10.

Summations of Single Terms and Successive Terms of Geometric Series

Number of pages: 2 Posted: 12 May 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 84 (423,220)

Abstract:

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Geometric series, summation, successive terms

11.

Factorial of Sum of Nonnegative Integers for Computing and Algorithms

Number of pages: 3 Posted: 17 Jun 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 76 (448,237)

Abstract:

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Algorithm, combinatorics, factorial, computation

12.

Sum of Summations of Annamalai’s Binomial Expansions

Number of pages: 4 Posted: 01 Jun 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 74 (454,883)

Abstract:

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computation, combinatorics, binomial coefficient, computer science

13.

Analysis of the Relationship between Integers and Factorial Functions

Number of pages: 4 Posted: 17 Jun 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 73 (458,403)

Abstract:

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Algorithm, combinatorics, factorial, computation

14.

A Binomial Expansion Equal to Multiple of 2 with Non-Negative Exponents

Number of pages: 3 Posted: 06 Jun 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 69 (472,308)

Abstract:

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binomial theorem, theoretical computer science, exponents, combinatorics

15.

Multiple Summations of a Geometric Series and Its Binomial Series

Number of pages: 2 Posted: 12 May 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 68 (475,889)

Abstract:

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binomial series, binomial coefficient, geometric series, multiple summations

16.

Factorials, Integers and Multinomial Coefficients and its Computing Techniques for Machine Learning and Cybersecurity

Number of pages: 7 Posted: 22 Aug 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 67 (479,528)

Abstract:

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algorithm, combinatorics, computation, multinomial coefficient

17.

Ascending and Descending Orders of Annamalai’s Binomial Coefficient

Number of pages: 4 Posted: 01 Jun 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 64 (490,996)

Abstract:

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combinatorics, binomial coefficient, combination, computation

18.

Theorems based on Annamalai’s Binomial Coefficient and Identity

Number of pages: 4 Posted: 02 Jun 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 61 (502,765)

Abstract:

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binomial theorem, factorial function, binomial coefficient

19.

Sum of the Summation of Binomial Expansions with Optimized Binomial Coefficient

Chinnaraji Annamalai. Sum of the Summation of Binomial Expansions with Optimized Binomial Coefficient. 2022. ffhal-03694065f
Number of pages: 4 Posted: 23 Jun 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 59 (510,819)

Abstract:

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combinatorics, binomial coefficient, computing, binomial expansion

20.

Combinatorial Techniques for Binomial Expansions with Multiples of 2

Chinnaraji Annamalai. Combinatorial Techniques for Binomial Expansions with Multiples of 2. 2022. ffhal-03694060f
Number of pages: 5 Posted: 23 Jun 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 59 (510,819)

Abstract:

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combinatorics, binomial coefficient, computing, binomial expansion

21.

Combinatorial Relation of Optimized Combination with Permutation

Number of pages: 2 Posted: 12 May 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 59 (510,819)

Abstract:

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combinatorics, counting technique, optimized combination

22.

Computation of Geometric Series in Different Ways

Chinnaraji Annamalai. Computation of Geometric Series in Different Ways. 2022. ffhal-03698759f
Number of pages: 3 Posted: 06 Jul 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 58 (514,916)

Abstract:

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computation, summation, geometric series

23.

Differentiation and Integration of Annamalai’s Binomial Expansion

Number of pages: 4 Posted: 02 Jun 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 56 (523,443)

Abstract:

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Differentiation, integration, binomial expansion, geometric series

24.

Factorial of Sum of Two Nonnegative Integers Is Equal to Multiple of the Product of Factorial of the Two Nonnegative Integers

Number of pages: 3 Posted: 17 Jun 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 54 (532,043)

Abstract:

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algorithm, combinatorics, factorial function, computation

25.

Extension of Binomial Series with Optimized Binomial Coefficient

Number of pages: 5 Posted: 17 Jun 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 53 (536,517)

Abstract:

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binomial expansion, geometric series, computation

26.

Computing Method for the Summation of Series of Binomial Coefficients

Chinnaraji Annamalai. Computing Method for the Summation of Series of Binomial Coefficients. 2022. ffhal-03698756f
Number of pages: 4 Posted: 06 Jul 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 47 (564,372)

Abstract:

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binomial coefficient, geometric series, summation

27.

Computing Method for Combinatorial Geometric Series and Binomial Expansion

Number of pages: 13 Posted: 25 Jul 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 39 (606,486)

Abstract:

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theoretical computer science, combinatorics, discrete mathematics

28.

Factorials, Integers, and Factorial Theorems for Computing and Cryptography

Number of pages: 3 Posted: 13 Sep 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 28 (675,480)

Abstract:

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algorithm, combinatorics, factorial, computation

29.

Computation for the Summation of Integers and Geometric Progression of Powers of Two

Number of pages: 3 Posted: 21 Jul 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 28 (675,480)

Abstract:

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computation, geometric series, powers of two, summation

30.

Computation for the Summation of Binomial Expansions and Geometric Series of Multiples of Powers of Two

Number of pages: 4 Posted: 20 Jul 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 26 (690,123)

Abstract:

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computation, combinatorics, geometric series, binomial coefficient

31.

Computation of Factorial and Multinomial Theorems for Machine Leaning and Cybersecurity

Number of pages: 7 Posted: 16 Sep 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 22 (721,479)

Abstract:

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algorithm, combinatorics, computation, multinomial

32.

Factorials, Integers and Mathematical and Binomial Techniques for Machine Learning and Cybersecurity

Number of pages: 6 Posted: 02 Aug 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 19 (746,430)

Abstract:

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algorithm, combinatorics, computation, prime number

33.

Differential Calculus for the Summation of Geometric Series with Binomial Expansions

Number of pages: 4 Posted: 27 Jul 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 18 (755,154)

Abstract:

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34.

Algorithmic Technique for Computation of Binomial Expansions and Geometric Series of Multiples of Powers of Two

Number of pages: 4 Posted: 23 Jul 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 17 (764,019)

Abstract:

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Computation, combinatorics, geometric series, binomial coefficient

35.

Multinomial Theorem on the Binomial Coefficients for Combinatorial Geometric Series

Number of pages: 4 Posted: 13 Sep 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 15 (782,180)

Abstract:

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computation, combinatorics, binomial expansion

36.

Computation Method for Combinatorial Geometric Series and its Applications

Number of pages: 15 Posted: 22 Aug 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 15 (782,180)

Abstract:

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computation, combinatorics, binomial theorem

37.

Combinatorial Geometric Series

HAL, Chinnaraji Annamalai. Combinatorial Geometric Series. 2022. ffhal-03723640f
Number of pages: 3 Posted: 04 Aug 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 14 (791,848)

Abstract:

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computation, combinatorics, binomial coefficient

38.

Computation of Geometric Series with Negative Exponents

Number of pages: 4 Posted: 22 Aug 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 12 (811,447)

Abstract:

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computation, derivative, geometric series, negative exponents

39.

Computation of Derivative of Geometric Series without Differentiation

Number of pages: 3 Posted: 22 Aug 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 11 (821,621)

Abstract:

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computation, derivative, geometric series

40.

New Idea to Compute the Geometric Series and its Derivative

Number of pages: 2 Posted: 16 Aug 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 11 (821,621)

Abstract:

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computation, geometric series, summation, derivative

41.

Factorials, Integers, Binomial Coefficient and Factorial Theorem

Number of pages: 2 Posted: 21 Sep 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 10 (831,996)

Abstract:

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Algorithm, combinatorics, factorial, computation

42.

Binomial Coefficients in Combinatorial Geometric Series and its Combinatorial Identities

Number of pages: 3 Posted: 30 Aug 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 10 (831,996)

Abstract:

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computation, combinatorics, binomial coefficient

43.

Binomial Coefficients of Combinatorial Geometric Series: System of Natural Numbers

Number of pages: 3 Posted: 11 Nov 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 9 (842,477)

Abstract:

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Computation, combinatorics, binomial coefficient

44.

Partition of Multinomial Coefficient

Number of pages: 4 Posted: 26 Oct 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 9 (842,477)

Abstract:

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Computation, combinatorics, binomial, multinomial, partition

45.

Binomial Coefficients and Identities in Combinatorial Geometric Series

Number of pages: 3 Posted: 14 Sep 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 9 (842,477)

Abstract:

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computation, combinatorics, binomial expansion

46.

Summation of Series of Binomial Coefficients

Number of pages: 2 Posted: 02 Nov 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 8 (853,157)

Abstract:

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computation, combinatorics, binomial coefficient

47.

Annamalai’s Binomial Expansion

Number of pages: 2 Posted: 30 Nov 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 7 (864,199)

Abstract:

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binomial expansion, combinatorics, computation

48.

Multinomial-based Factorial Theorem on the Binomial Coefficients for Combinatorial Geometric Series

Number of pages: 2 Posted: 20 Oct 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 7 (864,199)

Abstract:

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computation, combinatorics, binomial expansion

49.

Binomial Theorem on the Coefficients for Combinatorial Geometric Series

Number of pages: 3 Posted: 11 Oct 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 7 (864,199)

Abstract:

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Computation, combinatorics, binomial coefficient

50.

Lemma on Combinatorial Geometric Series with Binomial Coefficients

Number of pages: 3 Posted: 01 Dec 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 6 (875,408)

Abstract:

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computation, combinatorics, binomial coefficient

51.

Sum of Successive Partitions of Binomial Coefficient

Number of pages: 3 Posted: 17 Nov 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 6 (875,408)

Abstract:

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Computation, combinatorics, binomial coefficient, partition method

52.

Sum of Combinatorial Geometric Series

Number of pages: 3 Posted: 11 Nov 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 6 (875,408)

Abstract:

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Computation, combinatorics, binomial coefficient

53.

Computation and Analysis of Combinatorial Geometric Series and Binomial Series

Number of pages: 8 Posted: 31 Oct 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 6 (875,408)

Abstract:

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computation, binomial coefficient, binomial series, combinatorics

54.

Successive Partition Method for Binomial Coefficient in Combinatorial Geometric Series

Number of pages: 3 Posted: 25 Oct 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 6 (875,408)

Abstract:

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Computation, combinatorics, binomial coefficient, partition

55.

A Generalized Method for Proving the Theorem derived from the Binomial Coefficients in Combinatorial Geometric Series

Number of pages: 3 Posted: 19 Oct 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 6 (875,408)

Abstract:

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Computation, combinatorics, binomial coefficient

56.

Theorems on the Binomial Coefficients for Combinatorial Geometric Series

Number of pages: 3 Posted: 14 Oct 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 6 (875,408)

Abstract:

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Computation, combinatorics, binomial coefficient

57.

Combinatorial Theorems in Factorials with Multinomial Computation

Number of pages: 3 Posted: 16 Sep 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 6 (875,408)

Abstract:

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Algorithm, combinatorics, factorial, computation

58.

A Theorem on the Annamalai’s Binomial Identities

Number of pages: 4 Posted: 03 Aug 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 6 (875,408)

Abstract:

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computation, combinatorics, binomial theorem

59.

A Theorem on Binomial Series

Number of pages: 4 Posted: 02 Dec 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 5 (886,797)

Abstract:

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computation, binomial coefficient, binomial series, combinatorics

60.

Two Different and Equal Coefficients of Combinatorial Geometric Series

Number of pages: 3 Posted: 02 Dec 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 5 (886,797)

Abstract:

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computation, binomial coefficient, combinatorics

61.

Construction and Analysis of Binomial Coefficients

Number of pages: 3 Posted: 14 Nov 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 5 (886,797)

Abstract:

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Computation, combinatorics, combinatorial geometric series

62.

Real and Complex Numbers of Binomial Coefficients in Combinatorial Geometric Series

Number of pages: 3 Posted: 11 Nov 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 5 (886,797)

Abstract:

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Computation, combinatorics, binomial coefficient

63.

Abelian Group on the Binomial Coefficients of Combinatorial Geometric Series

Number of pages: 3 Posted: 02 Nov 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 5 (886,797)

Abstract:

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Computation, combinatorics, binomial coefficient, abelian group

64.

A Theorem on the Binomial Coefficients of Combinatorial Geometric Series and Some Solutions on Partitions of the Binomial Coefficients

Number of pages: 4 Posted: 28 Oct 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 5 (886,797)

Abstract:

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computation, combinatorics, binomial expansion

65.

A Theorem on Successive Partitions of Binomial Coefficient

Number of pages: 5 Posted: 19 Oct 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 5 (886,797)

Abstract:

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computation, combinatorics, binomial coefficient, partition method

66.

Scalar and Vector Space of Combinatorial Geometric Series

Number of pages: 4 Posted: 16 Nov 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 4 (898,790)

Abstract:

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combinatorics, binomial coefficient, scalar, vector space

67.

Combinatorial Geometric Series: Vector Space

Number of pages: 4 Posted: 11 Nov 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 4 (898,790)

Abstract:

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Combinatorics, binomial coefficient, abelian group, vector space

68.

Algorithmic Approach for Computation of Binomial Expansions

Number of pages: 3
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 3

Abstract:

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computation, binomial coefficient, combinatorics

69.

Construction of Novel Binomial Expansion

Number of pages: 2
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 1

Abstract:

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binomial coefficient, computation, combinatorics

70.

Novel Multinomial Expansion and Theorem

Number of pages: 2
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 1

Abstract:

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computation, binomial coefficient, combinatorics, multinomial

Other Papers (1)

Total Downloads: 3
1.

Alternative to the Binomial Series or Binomial Theorem

Number of pages: 3 Posted: 23 Nov 2022
Chinnaraji Annamalai
Indian Institute of Technology (IIT), Kharagpur - Vinod Gupta School of Management
Downloads 3

Abstract:

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computation, combinatorics, binomial coefficient