Genocchi number explained

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

(2 t)/(1+ e ^{t}) = \sum ^{inf ty} _{n =0} G _{n (t ^{n})/(n!)}

The first few Genocchi numbers are 0, 1, -1, 0, 1, 0, -3, 0, 17, see .

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

G _{n} =2 ( 1-2 ^{n)} B _{n.}

Combinatorial interpretations

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

t\tan \left( (t)/(2) \right) = \sum _{n\geq} ( -1 ) ^{n} G _{2n} (t ^{2n})/(( 2 n)!)

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 [2''n''&minus;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

• • Richard P. Stanley (1999). Enumerative Combinatorics, Volume 2, Exercise 5.8. Cambridge University Press.
• 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