Genocchi Number

A number given by the generating function

(1)

It satisfies , , and even coefficients are given by

			(2)
			(3)

where  is a Bernoulli number and  is an Euler polynomial.

The first few Genocchi numbers for , 4, ... are , 1, , 17, , 2073, ... (OEIS A001469).

The first few prime Genocchi numbers are  and 17, which occur for  and 8. There are no others with  (Weisstein, Mar. 6, 2004). D. Terr (pers. comm., Jun. 8, 2004) proved that these are in fact, the only prime Genocchi numbers.

REFERENCES:

Catalan, E. "Sur le calcul des Nombres de Bernoulli." C. R. Acad. Sci. Paris 58, 1105-1108, 1864.

Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 49, 1974.

Kreweras, G. "An Additive Generation for the Genocchi Numbers and Two of its Enumerative Meanings." Bull. Inst. Combin. Appl. 20, 99-103, 1997.

Kreweras, G. "Sur les permutations comptées par les nombres de Genocchi de 1-ière et 2-ième espèce." Europ. J. Comb. 18, 49-58, 1997.

Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684-760, 1973.

Sloane, N. J. A. Sequence A001469/M3041 in "The On-Line Encyclopedia of Integer Sequences."