Genocchi number

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

The first few Genocchi numbers are 1, 1, 0, 1, 0, 3, 0, 17 (sequence A036968 in the OEIS), see OEIS: A001469.

Properties

There are two cases for .

1.     from OEIS: A027641 / OEIS: A027642
= 1, -1, 0, 1, 0, -3 = OEIS: A036968, see OEIS: A224783
2.     from OEIS: A164555 / OEIS: A027642
= -1, -1, 0, 1, 0, -3 = OEIS: A226158(n+1). Generating function: .

OEIS: A226158 is an autosequence (a sequence whose inverse binomial transform is the signed sequence) of the first kind (its main diagonal is 0's = OEIS: A000004). An autosequence of the second kind has its main diagonal equal to the first upper diagonal multiplied by 2. Example: OEIS: A164555 / OEIS: A027642.

OEIS: A226158 is included in the family:

......11/20-1/401/20-17/8031/2
...0110-1030-170155
00230-50210-15301705

The rows are respectively OEIS: A198631(n) / OEIS: A006519(n+1), OEIS: A226158, and OEIS: A243868.

A row is 0 followed by n (positive) multiplied by the preceding row. The sequences are alternatively of the second and the first kind.

  • It has been proved that 3 and 17 are the only prime Genocchi numbers.

Combinatorial interpretations

The exponential generating function for the signed even Genocchi numbers (1)nG2n is

They enumerate the following objects:

  • Permutations in S2n1 with descents after the even numbers and ascents after the odd numbers.
  • Permutations π in S2n2 with 1  π(2i1)  2n2i and 2n2i  π(2i)  2n2.
  • Pairs (a1,,an1) and (b1,,bn1) such that ai and bi are between 1 and i and every k between 1 and n1 occurs at least once among the ai's and bi's.
  • Reverse alternating permutations a1 < a2 > a3 < a4 >>a2n1 of [2n1] whose inversion table has only even entries.

See also

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
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