Genocchi number

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

${\displaystyle {\frac {2t}{1+e^{t}}}=\sum _{n=0}^{\infty }G_{n}{\frac {t^{n}}{n!}}}$

The first few Genocchi numbers are 0, 1, −1, 0, 1, 0, −3, 0, 17 (sequence A226158 in the OEIS), see OEIS: A001469.

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

where ${\displaystyle \zeta (1-n)}$ is the Riemann zeta function.

Properties

• The generating function definition of the Genocchi numbers implies that they are rational numbers. In fact, 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

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

Combinatorial interpretations

The exponential generating function for the signed even Genocchi numbers (−1)ⁿG_2n is

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

They enumerate the following objects:

• Permutations in 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.

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