Genocchi number

In mathematics, the Genocchi numbers Gn, 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 .

Properties

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

There are two cases for $G_n$.

1. $B_1 = -1/2$     from /
$G_{n_{1}}$ = 1, -1, 0, 1, 0, -3 = , see
2. $B_1 = 1/2$     from /
$G_{n_{2}}$ = -1, -1, 0, 1, 0, -3 = . Generating function: $\frac{-2}{1+e^{-t}}$ .

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

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 (n) / (n+1), −, and .

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)nG2n is

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

They enumerate the following objects:

• Permutations π in S2n−2 with 1 ≤ π(2i−1) ≤ 2n−2i and 2n−2i ≤ π(2i) ≤ 2n−2.
• Pairs (a1,…,an−1) and (b1,…,bn−1) such that ai and bi are between 1 and i and every k between 1 and n−1 occurs at least once among the ai's and bi's.