A064625 -id:A064625 - cp's OEIS frontend

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.

1, 1, 7, 145, 6631, 566641, 81184327, 18070338385, 5905039303591, 2711929990866481, 1690633724369840647, 1390752644563701636625, 1474612871875198657851751, 1975728790062794178772769521

Offset: 0

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

			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]

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).

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.

Cf. A001469, A064625.

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-François Alcover, May 23 2013 *)

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

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]

A065748 Triangle of Gandhi polynomial coefficients.

Original entry on oeis.org

1, 1, 4, 6, 4, 15, 88, 220, 304, 250, 120, 28, 1025, 7308, 23234, 43420, 52880, 43880, 25088, 9680, 2340, 280, 209135, 1691024, 6237520, 13911400, 20956610, 22549360, 17853780, 10541440, 4639740, 1498280, 341000, 49920, 3640, 100482849

Offset: 1

Mike Domaratzki (mdomaratzki(AT)alumni.uwaterloo.ca), Nov 16 2001

First column is A064625.

			Triangle starts
1;
1,4,6,4;
15,88,220,304,250,120,28;
1025,...

Michael Domaratzki, Combinatorial Interpretations of a Generalization of the Genocchi Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.6.
D. Dumont, Sur une conjecture de Gandhi concernant les nombres de Genocchi, (in French), Discrete Mathematics 1 (1972) 321-327.
D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.
D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318. (Annotated scanned copy)

Cf. A064625, A036970

Let B(X, n) = X^4 (B(X+1, n-1) - B(X, n-1)), B(X, 1) = X^4; then the (i, j)-th entry in the table is the coefficient of X^(5+j) in B(X, i).

A065756 Generalization of the Genocchi numbers given by the Gandhi polynomials A(n+1,r) = r^5 A(n, r + 1) - (r - 1)^5 A(n, r); A(1,r) = r^5 - (r-1)^5.

Original entry on oeis.org

1, 1, 31, 6721, 5850271, 15060446401, 94396946822431, 1258620297379341121, 32323181593821704288671, 1481630482369728860007652801, 114129022540066183425609121804831

Offset: 1

Mike Domaratzki (mdomaratzki(AT)alumni.uwaterloo.ca), Nov 17 2001

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).
Michael Domaratzki, Combinatorial Interpretations of a Generalization of the Genocchi Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.6.
D. Dumont, Sur une conjecture de Gandhi concernant les nombres de Genocchi, (in French), Discrete Mathematics 1 (1972) 321-327.
D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.
D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318. (Annotated scanned copy)

Cf. A001469, A065755, A065757, A064624, A064625.

a[n_ /; n >= 0, r_ /; r >= 0] := a[n, r] = r^5*a[n-1, r+1]-(r-1)^5*a[n-1, r]; a[1, r_ /; r >= 0] := r^5-(r-1)^5; a[, ] = 1; a[n_] := a[n-1, 1]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, May 23 2013 *)

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

A065754 Second column of A065748.

Original entry on oeis.org

4, 88, 7308, 1691024, 889922900, 927968206536, 1736907362701852, 5418843464230116352, 26603238870832186065636, 196293745325282121998886200, 2096654942154151785036724164524

Offset: 1

Mike Domaratzki (mdomaratzki(AT)alumni.uwaterloo.ca), Nov 17 2001

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

Cf. A065748, A064625

cp's OEIS Frontend

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.

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A065748 Triangle of Gandhi polynomial coefficients.

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A065756 Generalization of the Genocchi numbers given by the Gandhi polynomials A(n+1,r) = r^5 A(n, r + 1) - (r - 1)^5 A(n, r); A(1,r) = r^5 - (r-1)^5.

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A065754 Second column of A065748.

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