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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
1, 1, 7, 145, 6631, 566641, 81184327, 18070338385, 5905039303591, 2711929990866481, 1690633724369840647, 1390752644563701636625, 1474612871875198657851751, 1975728790062794178772769521 (list; graph; refs; listen; history; text; internal format)
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 of Kingston (Kingston, Canada).

LINKS

Table of n, a(n) for n=0..13.

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

EXAMPLE

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

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]

MATHEMATICA

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 *)

PROG

(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)}

CROSSREFS

Cf. A001469, A064625.

Sequence in context: A012826 A232363 A102059 * A342113 A211108 A251668

Adjacent sequences:  A064621 A064622 A064623 * A064625 A064626 A064627

KEYWORD

easy,nonn

AUTHOR

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

STATUS

approved

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Last modified July 29 22:13 EDT 2021. Contains 346346 sequences. (Running on oeis4.)