Genocchi number

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



\frac{2t}{1+e^{t}}=\sum_{n=0}^\infty G_n\frac{t^n}{n!} $$

The first few Genocchi numbers are 0, 1, &minus;1, 0, 1, 0, &minus;3, 0, 17, see.

Properties

 * The generating function definition of the Genocchi numbers implies that they are rational numbers. In fact, G2n+1 = 0 for n &ge; 1 and (&minus;1)nG2n is an odd positive integer.
 * Genocchi numbers Gn are related to Bernoulli numbers Bn by the formula



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

Combinatorial interpretations
The exponential generating function for the signed even Genocchi numbers (&minus;1)nG2n is

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 S2n&minus;1 with descents after the even numbers and ascents after the odd numbers.
 * Permutations &pi; in S2n&minus;2 with 1 &le; &pi;(2i&minus;1) &le; 2n&minus;2i and 2n&minus;2i &le; &pi;(2i) &le; 2n&minus;2.
 * Pairs (a1,...,an&minus;1) and (b1,...,bn&minus;1) such that ai and bi are between 1 and i and every k between 1 and n&minus;1 occurs at least once among the ai's and bi's.
 * Reverse alternating permutations a1 < a2 > a3 < a4 >...>a2n&minus;1 of [2n&minus;1] 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 (depending on how primes are defined). It has been proven that no other primes occur in the sequence