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Genocchi Number

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A number given by the generating function

(2t)/(e^t+1)=sum_(n=1)^inftyG_n(t^n)/(n!).
(1)

It satisfies G_1=1, G_3=G_5=G_7=...=0, and even coefficients are given by

G_(2n)=2(1-2^(2n))B_(2n)
(2)
=2nE_(2n-1)(0),
(3)

where B_n is a Bernoulli number and E_n(x) is an Euler polynomial.

The first few Genocchi numbers for n=2, 4, ... are -1, 1, -3, 17, -155, 2073, ... (OEIS A001469).

The first few prime Genocchi numbers are -3 and 17, which occur for n=6 and 8. There are no others with n<10^5 (Weisstein, Mar. 6, 2004). D. Terr (pers. comm., Jun. 8, 2004) proved that these are in fact, the only prime Genocchi numbers.

See also

Bernoulli Number, Euler Polynomial, Integer Sequence Primes

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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."

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Genocchi Number

Cite this as:

Weisstein, Eric W. "Genocchi Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GenocchiNumber.html

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