Genocchi number

Properties

The generating function definition of the Genocchi numbers implies that they are rational numbers. In fact, G_2n+1 = 0 for n ≥ 1 and (−1)ⁿG_2n is an odd positive integer.

Genocchi numbers G_n are related to Bernoulli numbers B_n by the formula

G n = 2 ( 1 − 2 n ) B n .

There are two cases for G n .

1. B 1 = − 1 / 2     from  A027641 /  A027642 G n 1 = 1, -1, 0, 1, 0, -3 =  A036968, see  A2247832. B 1 = 1 / 2     from  A164555 /  A027642 G n 2 = -1, -1, 0, 1, 0, -3 =  A226158(n+1). Generating function: − 2 1 + e − t .

 A226158 is an autosequence (a sequence whose inverse binomial transform is the signed sequence) of the first kind (its main diagonal is 0's =  A000004). An autosequence of the second kind has its main diagonal equal to the first upper diagonal multiplied by 2. Example:  A164555 /  A027642.

− A226158 is included in the family:

The rows are respectively  A198631(n) /  A006519(n+1), − A226158, and  A243868.

A row is 0 followed by n (positive) multiplied by the preceding row. The sequences are alternatively of the second and the first kind.

It has been proved that −3 and 17 are the only prime Genocchi numbers.

Combinatorial interpretations

The exponential generating function for the signed even Genocchi numbers (−1)ⁿG_2n is

t tan ⁡ ( t 2 ) = ∑ n ≥ 1 ( − 1 ) n G 2 n t 2 n ( 2 n ) !

They enumerate the following objects:

Permutations in S_2n−1 with descents after the even numbers and ascents after the odd numbers.

Permutations π in S_2n−2 with 1 ≤ π(2i−1) ≤ 2n−2i and 2n−2i ≤ π(2i) ≤ 2n−2.

Pairs (a₁,…,a_n−1) and (b₁,…,b_n−1) such that a_i and b_i are between 1 and i and every k between 1 and n−1 occurs at least once among the a_i's and b_i's.

Reverse alternating permutations a₁ < a₂ > a₃ < a₄ >…>a_2n−1 of [2n−1] whose inversion table has only even entries.