Genocchi number

In mathematics, the Genocchi numbers G_n, 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  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

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

There are two cases for $G_n$.

1. $B_1 = -1/2$     from  A027641 /  A027642

$G_{n_{1}}$ = 1, -1, 0, 1, 0, -3 =  A036968, see  A224783

2. $B_1 = 1/2$     from  A164555 /  A027642

$G_{n_{2}}$ = -1, -1, 0, 1, 0, -3 =  A226158(n+1). Generating function: $\frac{-2}{1+e^{-t}}$ .

 A226158 is an autosequence (a sequence whose inverse binomial transform is the signed sequence) of the first kind (its main diagonal is 0's =  A000004). An autosequence of the second kind has its main diagonal equal to the first upper diagonal multiplied by 2. Example:  A164555 /  A027642.

− A226158 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  A198631(n) /  A006519(n+1), − A226158, and  A243868.

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)ⁿG_2n is

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

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.

See also

• Euler number

References

• Weisstein, Eric W., "Genocchi Number", MathWorld.
• Richard P. Stanley (1999). Enumerative Combinatorics, Volume 2, Exercise 5.8. Cambridge University Press. ISBN 0-521-56069-1
• Some Results for the Apostol-Genocchi Polynomials of Higher Order, Hassan Jolany, Hesam Sharifi and R. Eizadi Alikelaye, Bull. Malays. Math. Sci. Soc. (2) 36(2) (2013), 465–479
• Gérard Viennot, Interprétations combinatoires des nombres d'Euler et de Genocchi, Seminaire de Théorie des Nombres de Bordeaux, Volume 11 (1981-1982)
• Serkan Araci, Mehmet Acikgoz, Erdoğan Şen, Some New Identities of Genocchi Numbers and Polynomials