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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.

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[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

Mathematics Encyclopedia

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