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In mathematics, the Genocchi numbers Gn, 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 OEIS A001469.

Properties

The generating function definition of the Genocchi numbers implies that they are rational numbers. In fact, G2n+1 = 0 for n ≥ 1 and (−1)nG2n is an odd positive integer.

Genocchi numbers Gn are related to Bernoulli numbers Bn by the formula

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

There are two cases for G_n.

1. $$\fB_1 = -1/2$$ from OEIS A027641 / OEIS A027642

$$\fG_{n_{1}} = 1, -1, 0, 1, 0, -3$$ = OEIS A036968, see OEIS A224783

2. $$\fB_1 = 1/2$$ from OEIS A164555 / OEIS A027642

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

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

−OEIS 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 OEIS A198631(n) / OEIS A006519(n+1), −OEIS A226158, and OEIS 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)nG2n 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 S2n−1 with descents after the even numbers and ascents after the odd numbers.
• Permutations π in S2n−2 with 1 ≤ π(2i−1) ≤ 2n−2i and 2n−2i ≤ π(2i) ≤ 2n−2.
• Pairs (a1,…,an−1) and (b1,…,bn−1) such that ai and bi are between 1 and i and every k between 1 and n−1 occurs at least once among the ai's and bi's.
• Reverse alternating permutations a1 < a2 > a3 < a4 >…>a2n−1 of [2n−1] whose inversion table has only even entries.

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[1]

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