Genocchi number

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

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

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

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

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

1. ${\displaystyle B_{1}=-1/2}$     from  A027641 /  A027642

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

2. $B_{1}=1/2$     from  A164555 /  A027642

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

 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

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

They enumerate the following objects:

• Permutations in 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