Genocchi number

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

\( {\displaystyle {\frac {-2t}{1+e^{-t}}}=\sum _{n=0}^{\infty }G_{n}{\frac {t^{n}}{n!}}} \)

The first few Genocchi numbers are 0, −1, −1, 0, 1, 0, −3, 0, 17 (sequence A226158 in the 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 G_n are related to Bernoulli numbers B_n by the formula

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

Combinatorial interpretations

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

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

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.