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

Author

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

Keywords

• easy
• nonn

Examples

			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]
		

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

• 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.

Crossrefs

Cf. A001469, A064624.

Programs

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 *)

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)}

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]