- 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
There are two cases for .
- 1. from A027641 / A027642
- 2. from A164555 / A027642
- = -1, -1, 0, 1, 0, -3 = A226158(n+1). Generating function: .
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:
A row is 0 followed by n (positive) multiplied by the preceding row. The sequences are alternatively of the second and the first kind.
The exponential generating function for the signed even Genocchi numbers (−1)nG2n is
They enumerate the following objects:
- 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.
- 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