### Combinatorial numbers and Egorychev method

These are binomial coefficients, Stirling numbers, Catalan numbers, Harmonic numbers, Eulerian numbers and Bernoulli numbers, as in the book Integral representation and the Computation of Combinatorial Sums by G. P. Egorychev.

The collection including a list of all identities proved is available here (part 1 and 2, formal power series and residue operators):

and here (part 3, complex variables) This includes a number of identities from H.W. Gould's book Combinatorial identities as well as L. Saalschütz's book on Bernoulli numbers.

We now have a new section as of March 2023 "computer search" which includes hypergeometric identities that can prove challenging even for computer algebra systems, such as

As of January 2024 "computer search" continues with rare identities such as (the 42 identity)

also featuring Narayana and Catalan numbers. These computer identities have ranges of validity (boundary conditions) which are documented in the text.

Here are the slides from Hosam Mahmoud's talk on Egorychev method at Catholic University on November 9, 2022: History and examples of Egorychev method.

The 2023 paper "Egorychev method: a hidden treasure" by Riedel and Mahmoud is at the following Springer link.

A MathJax version of this list can be found here.

This is the list of identities in the above document, including links to the posts where they first appeared.

# Part 1 and 2

### Egorychev method in formal power series

1. MSE 2384932
2. MSE 2472978
3. MSE 2719320
4. MSE 2830860
5. MSE 2904333
6. MSE 2950043
7. MSE 3049572
8. MSE 3051713
9. MSE 3068381
10. MSE 3138710
11. MSE 3196998
12. MSE 3245099
13. MSE 3260307
14. MSE 3285142
15. MSE 3333597
16. MSE 3342361
17. MSE 3383557
18. MSE 3441855
19. MSE 3577193
20. MSE 3583191
21. MSE 3592240
22. MSE 3604802
23. MSE 3619182
24. MSE 3638162
25. MSE 3661349
26. MSE 3706767
27. MSE 3737197
28. MSE 3825092
29. MSE 3845061
30. MSE 3885278
31. MSE 3559223
32. MSE 3926409
33. MSE 3942039
34. MSE 3956698
35. MSE 3993530
36. MSE 4008277
37. MSE 4031272 (Li Shanlan identity)
38. MSE 4034224 (Eulerian numbers, Stirling numbers first and second kind)
39. MSE 4037172 (Eulerian numbers, associated Stirling numbers first and second kind)
40. MSE 4037946 (Eulerian numbers, associated Stirling numbers first and second kind)
41. MSE 4055292
42. MSE 4054024
43. MSE 4084763
44. MSE 4095795
45. MSE 4098492
46. MSE 4127695
47. MSE 4131219
48. MSE 4139722
49. MSE 4192271
50. MSE 4212878
51. A different obstacle from Concrete Mathematics by Knuth, Graham, and Patashnik
52. MO 291738 (not quite the same as previous)
53. A Stirling number identity by Gould
54. A Stirling number identity by Gould II
55. Schläfli's identity for Stirling numbers
56. Stirling numbers and Faulhaber's formula
57. Stirling numbers and binomial coefficient
58. Stirling numbers and a double binomial coefficient
59. Stirling numbers and a double binomial coefficient II
60. Stirling numbers and Bernoulli polynomials
61. Central binomial coefficients and Stirling numbers
62. Single variable monomial and two binomial coefficients
63. Use of an Iverson bracket
64. Use of an Iverson bracket II
65. Use of an Iverson bracket III
66. Basic example
67. Basic example continued
68. An identity by Erik Sparre Andersen
69. Very basic example
70. An identity by Karl Goldberg
71. Sum producing a square root
72. Sum producing a square root II
73. Use of an Iverson bracket IV
74. Binomial coefficient manipulation
75. Four binomial sums
76. Power term and two binomial coefficients
77. Use of an Iverson bracket V
78. Use of an Iverson bracket VI
79. Appearance of the constants three and five
80. Generating function of a binomial term
81. Double square root
82. Central Delannoy numbers
83. A case of factorization
84. Two identities due to Grosswald
85. Appearance of the constant three
86. Very basic example
87. Very basic example II
88. Nested square root
89. Harmonic numbers and a squared binomial coefficient
90. Harmonic numbers and a double binomial coefficient
91. Two instances of a Harmonic number
92. Legendre Polynomials
93. Legendre Polynomials and a square root
94. Legendre Polynomials and a double square root
95. MSE 4304623
96. Legendre Polynomials, trigonometric terms and a contour integral
97. Sum independent of a variable
98. Polynomial in three variables
99. An identity by Van der Corput
100. MSE 4316037 Logarithm, binomial coefficient and harmonic numbers
101. An identity credited to Chung
102. MSE 4317353 A Catalan number convolution
103. Odd index binomial coefficients
104. MSE 4325592 A sum of inverse binomial coefficients
105. Inverted sum index
106. MSE 4351714 A Catalan number recurrence
107. An identity by Graham and Riordan
108. Square root term
109. An identity by Machover and Gould
110. Moriarty identity by H.T.Davis et al.
111. Inverse Moriarty identity by Marcia Ascher.
112. Moriarty identity by Egorychev.
113. MSE 4462359 Two binomial coefficients
114. Polynomial identity
115. Polynomial identity II
116. Worpitzky-Nielsen series
117. MSE 4517120 A sum of inverse binomial coefficients
118. MSE 4520057 Symmetric Bernoulli number identity
119. Polynomial identity III
120. Polynomial identity IV
121. MSE 4540192 Symmetry in a simple proof
122. Free functional term
123. MSE 4547110 Inverse central binomial coefficient
124. MSE 1402886 Inverse binomial coefficient
125. MSE 4552694 A pair of binomial transforms
126. Polynomial with inverse binomial coefficients
127. Harmonic numbers with inverse binomial coefficients
128. Simon's identity
129. Identity from Abramowitz and Stegun / Schläfli's formula
130. Bernoulli / Stirling number identity identity
131. Recurrence relation from DLMF
132. Bernoulli, Fibonacci and Lucas numbers
133. Bernoulli / exponential convolution
134. Bernoulli identity by Munch
135. Bernoulli identity by Kronecker
136. Computing Bernoulli numbers

### From the Saalschütz text

• Bernoulli numbers I
• Bernoulli numbers II
• Bernoulli numbers III
• Bernoulli numbers IV
• Bernoulli and Eulerian numbers I
• Bernoulli and Eulerian numbers II
• Bernoulli numbers V
• Bernoulli numbers and roots of unity

### Computer search II

1. MSE 4902676: Inverse central binomial coefficient

2. MSE 4906245: Another inverse binomial coefficient

3. Quadruple hypergeometric with an inverse binomial coefficient
4. Quadruple hypergeometric with two inverse binomial coefficients
5. MSE 4949815: Inverse binomial coefficient with harmonic numbers

6. MSE 4960823: Binomial coefficient scaled power sum

7. MSE 4964157: Two central binomial coefficients and a fractional term

# Part 3

### Post Scriptum section

1. A trigonometric sum

2. A class of polynomials similar to Fibonacci and Lucas Polynomials

3. Partial row sums in Pascal's triangle

4. The tree function and Eulerian numbers of the second order

5. A Stirling set number generating function and Eulerian numbers of the second order

A Stirling cycle number generating function and Eulerian numbers of the second order (II)

6. Another case of factorization

7. An additional case of factorization

8. Contours and a binomial square root

9. A careful examination of contours

10. Stirling numbers, Bernoulli numbers and Catalan numbers from Concrete Mathematics by Graham, Knuth and Patashnik

11. Transforming an OGF into an EGF

12. Stirling numbers of the first and second kind

13. An identity by Carlitz

14. Logarithm of the Catalan number OGF

15. A Bernoulli / Stirling number identity

16. Formal power series vs contour integration