A064625 Generalization of the Genocchi numbers. Generated by the Gandhi polynomials A(n+1,r) = r^4 A(n,r+1) - (r-1)^4 A(n,r); A(1,r) = r^4 - (r-1)^4.

1, 1, 15, 1025, 209135, 100482849, 97657699279, 172687606607425, 513828770061202095, 2422699282016359575905, 17259669919850500726265231, 178741720937382151333667162241, 2605965447000176066894638515610735

Offset

0,3

References

M. Domaratzki, A Generalization of the Genocchi Numbers with Applications to Enumeration of Finite Automata. Technical Report 2001-449, Department of Computing and Information Science, Queen's University at Kingston (Kingston, Canada).

Links

Table of n, a(n) for n=0..12.

M. Domaratzki, A Generalization of the Genocchi Numbers with Applications to ...

Michael Domaratzki, Combinatorial Interpretations of a Generalization of the Genocchi Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.6.

Formula

a(n) = A(n-1, 1) for the above Gandhi polynomials.

O.g.f.: Sum_{n>=0} n!^4 * x^n / Product_{k=1..n} (1 + k^4*x). [From Paul D. Hanna, Jul 21 2011]

Example

O.g.f.: A(x) = 1 + x + 15*x^2 + 1025*x^3 + 209135*x^4 + 100482849*x^5 +...
where A(x) = 1 + x/(1+x) + 2!^4*x^2/((1+x)*(1+16*x)) + 3!^4*x^3/((1+x)*(1+16*x)*(1+81*x)) + 4!^4*x^4/((1+x)*(1+16*x)*(1+81*x)*(1+256*x)) +... [From Paul D. Hanna, Jul 21 2011]

Mathematica

a[n_ /; n >= 0, r_ /; r >= 0] := a[n, r] = r^4*a[n-1, r+1]-(r-1)^4*a[n-1, r]; a[1, r_ /; r >= 0] := r^4-(r-1)^4; a[_, _] = 1; a[n_] := a[n-1, 1]; Table[a[n], {n, 0, 12}] (* Jean-François Alcover, May 23 2013 *)

Prog

(PARI) {a(n)=polcoeff(sum(m=0, n, m!^4*x^m/prod(k=1, m, 1+k^4*x+x*O(x^n))), n)}

Crossrefs

Cf. A001469, A064624.

Sequence in context: A196569 A019282 A082158 * A241331 A205602 A370735

Adjacent sequences: A064622 A064623 A064624 * A064626 A064627 A064628

Keyword

easy,nonn

Author

Mike Domaratzki (mdomaratzki(AT)alumni.uwaterloo.ca), Sep 28 2001

Status

approved