.

In mathematics, the Genocchi numbers G_n, named after Angelo Genocchi, are a sequence of integers that satisfy the relation

\( \frac{2t}{e^t+1}=\sum_{n=1}^{\infty} G_n\frac{t^n}{n!} \)

The first few Genocchi numbers are 1, −1, 0, 1, 0, −3, 0, 17 (sequence A036968 in OEIS), see OEIS A001469.

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 Gn are related to Bernoulli numbers Bn by the formula

\( \f G_{n}=2 \,(1-2^n) \,B_n. \)

There are two cases for G_n.

1. \( \fB_1 = -1/2 \) from OEIS A027641 / OEIS A027642

\( \fG_{n_{1}} = 1, -1, 0, 1, 0, -3 \) = OEIS A036968, see OEIS A224783

2. \( \fB_1 = 1/2 \) from OEIS A164555 / OEIS A027642

\( \f G_{n_{2}} = -1, -1, 0, 1, 0, -3 \) = OEIS A226158(n+1). Generating function: \( \frac{-2}{1+e^{-t}} \) .

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

−OEIS A226158 is included in the family:
... ... 1 1/2 0 -1/4 0 1/2 0 -17/8 0 31/2
... 0 1 1 0 -1 0 3 0 -17 0 155
0 0 2 3 0 -5 0 21 0 -153 0 1705

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

\( t\tan(\frac{t}{2})=\sum_{n\geq 1} (-1)^n G_{2n}\frac{t^{2n}}{(2n)!} \)

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.