# Genocchi number

(Redirected from Genocchi prime)

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

${\displaystyle {\frac {2t}{e^{t}+1}}=\sum _{n=1}^{\infty }G_{n}{\frac {t^{n}}{n!}}}$

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

## Properties

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

There are two cases for ${\displaystyle G_{n}}$.

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

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

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

They enumerate the following objects:

• Permutations in S2n−1 with descents after the even numbers and ascents after the odd numbers.
• 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.
• Reverse alternating permutations a1 < a2 > a3 < a4 >…>a2n−1 of [2n−1] whose inversion table has only even entries.