A064625 - OEIS

%I #15 Jun 22 2013 13:16:27

%S 1,1,15,1025,209135,100482849,97657699279,172687606607425,

%T 513828770061202095,2422699282016359575905,17259669919850500726265231,

%U 178741720937382151333667162241,2605965447000176066894638515610735

%N 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.

%D 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).

%H M. Domaratzki, <a href="http://www.cs.queensu.ca/TechReports/Reports/2001-449.ps">A Generalization of the Genocchi Numbers with Applications to ...</a>

%H Michael Domaratzki, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Domaratzki/doma23.html">Combinatorial Interpretations of a Generalization of the Genocchi Numbers</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.6.

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

%F 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]

%e O.g.f.: A(x) = 1 + x + 15*x^2 + 1025*x^3 + 209135*x^4 + 100482849*x^5 +...

%e 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]

%t 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 *)

%o (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)}

%Y Cf. A001469, A064624.

%K easy,nonn

%O 0,3

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