Genocchi number

In mathematics, the Genocchi numbers G_n, named after Angelo Genocchi, are a sequence of integers that satisfy the relation
The first few Genocchi numbers are 0, 1, −1, 0, 1, 0, −3, 0, 17, see.

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 ⁿG_2n is an odd positive integer.
Genocchi numbers G_n are related to Bernoulli numbers B_n by the formula
Combinatorial interpretations

The exponential generating function for the signed even Genocchi numbers ⁿG_2n is
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 ≤ 2n−2i and 2n−2i ≤ π ≤ 2n−2.
Pairs and 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 whose inversion table has only even entries.
Primes

The only known prime numbers which occur in the Genocchi sequence are 17, at n = 8, and −3, at n = 6. It has been proven that no other primes occur in the sequence

Genocchi number

Properties

Combinatorial interpretations

Primes