# Table for Combinatorial Numbers and Associated Identities

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Table for Combinatorial Numbers and Associated Identities Table 2 From the seven unpublished manuscripts of H W Gould Edited and Compiled by Jocelyn Quaintance May 3 2010 1 Bernoulli Numbers Bn Remark 1 1 Throughout this chapter we assume n and p are nonnegative integers We assume x is a real or complex number Furthermore for any real x we let x denote the floor of x 1 1 Generating Function Definition of Bn X Bk k 0 1 2 xk k x ex 1 x 2 Alternative Definition for Bn p X p 1 p 1 k n Bk k p 1 k k 0 k 0 n 1 X 1 3 1 1 p 1 n 1 1 2 Explicit Formulas of Bn Bp p X j 0 1 j X j 1 k 0 k 1 j k kp n 1 k k X 2 X 1 k 1 n 1 X 1 n k k Bn B where Bk k 1 k j n n 1 k 1 k 1 k k j 1 j j j 0 1 1 3 1 4 n X n k n 1 Bk k Bn n 1 k 1 n k k 0 1 3 1 k 1 5 Alternative Formulations of Equation 1 3 Bn n X 1 k k 0 k 1 n Bk k 1 6 j n n X X 1 k X j 1 k n 1 j k n Bn Aj n where Aj n 1 k k 1 n k j 0 k 0 k 0 Bn 1 n n X Ak n k 1 Bn n X 1 k k 1 1 4 1 k 1 n n 1 k 1 Ak n n 1 nk n 1 1 7 1 8 n 1 1 9 Vandiver s Formulas for Bn Remark 1 2 The formulas in this section are the work of H S Vandiver The pertinent papers are On generalizations of the numbers of Bernoulli and Euler Proc of the National Academy of Sciences Vol 23 1937 pp 555 559 also see Proc of the National Academy of Sciences Vol 25 1939 pp 197 201 and Explicit expressions for generalized Bernoulli numbers Duke Math Journal Vol 8 No 3 Sept 1941 pp 575 584 n X k n 1 1 X n j Bn 1 k 1 k 1 j 0 k 0 k k 1 n X 1 X n k n 1 Bn 1 j k 1 k 1 j 0 k 0 1 10 n 1 k n 1 X n n 1 X n k Bn 1 1 j n 1 k 0 k 1 k 1 j 0 n 2 1 11 n 1 1 12 1 5 Properties of Bn 1 5 1 Recursive Relation n X n Bk 1 n Bn k k 0 1 5 2 1 13 Parity Properties B2n 2n X 1 k k 0 2n 1 X B2n