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A064624 Generalization of the Genocchi numbers given by the Gandhi polynomials A(n+1,r) = r^3 A(n, r + 1) - (r - 1)^3 A(n, r); A(1,r) = r^3 - (r-1)^3. 4

%I

%S 1,1,7,145,6631,566641,81184327,18070338385,5905039303591,

%T 2711929990866481,1690633724369840647,1390752644563701636625,

%U 1474612871875198657851751,1975728790062794178772769521

%N Generalization of the Genocchi numbers given by the Gandhi polynomials A(n+1,r) = r^3 A(n, r + 1) - (r - 1)^3 A(n, r); A(1,r) = r^3 - (r-1)^3.

%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 of 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!^3 * x^n / Product_{k=1..n} (1 + k^3*x). [From Paul D. Hanna, Jul 21 2011]

%e O.g.f.: A(x) = 1 + x + 7*x^2 + 145*x^3 + 6631*x^4 + 566641*x^5 +...

%e where A(x) = 1 + x/(1+x) + 2!^3*x^2/((1+x)*(1+8*x)) + 3!^3*x^3/((1+x)*(1+8*x)*(1+27*x)) + 4!^3*x^4/((1+x)*(1+8*x)*(1+27*x)*(1+64*x)) +... [From Paul D. Hanna, Jul 21 2011]

%t a[n_ /; n >= 0, r_ /; r >= 0] := a[n, r] = r^3*a[n-1, r+1] - (r-1)^3*a[n-1, r]; a[1, r_ /; r >= 0] := r^3-(r-1)^3; a[_, _] = 1; a[n_] := a[n-1, 1]; Table[a[n], {n, 0, 13}] (* _Jean-Fran├žois Alcover_, May 23 2013 *)

%o (PARI) {a(n)=polcoeff(sum(m=0,n,m!^3*x^m/prod(k=1,m,1+k^3*x+x*O(x^n))),n)}

%Y Cf. A001469, A064625.

%K easy,nonn

%O 0,3

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

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Last modified September 26 13:25 EDT 2021. Contains 347668 sequences. (Running on oeis4.)