In mathematics, the Genocchi numbers Gn, named after Angelo Genocchi, are a sequence of integers that satisfy the relation
2 t 1 + e t = ∑ n = 0 ∞ G n t n n ! {\displaystyle {\frac {2t}{1+e^{t}}}=\sum _{n=0}^{\infty }G_{n}{\frac {t^{n}}{n!}}}
The first few Genocchi numbers are 0, 1, −1, 0, 1, 0, −3, 0, 17 (sequence A226158 in the OEIS), see OEIS: A001469.
G n = 2 ( 1 − 2 n ) ( − n ζ ( 1 − n ) ) , {\displaystyle G_{n}=2(1-2^{n})(-n\zeta (1-n)),}
where ζ ( 1 − n ) {\displaystyle \zeta (1-n)} is the Riemann zeta function.
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
G n = 2 ( 1 − 2 n ) B n . {\displaystyle G_{n}=2\,(1-2^{n})\,B_{n}.}
Combinatorial interpretations The exponential generating function for the signed even Genocchi numbers (−1)nG2n is
t tan ( t 2 ) = ∑ n ≥ 1 ( − 1 ) n G 2 n t 2 n ( 2 n ) ! {\displaystyle t\tan \left({\frac {t}{2}}\right)=\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.
Primes The only known prime numbers which occur in the Genocchi sequence are 17, at n = 8, and −3, at n = 6 (depending on how primes are defined). It has been proven that no other primes occur in the sequence
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 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
