A064624 -id:A064624 - cp's OEIS frontend

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

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

			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]

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).

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, A064624.

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 *)

{a(n)=polcoeff(sum(m=0,n,m!^4*x^m/prod(k=1,m,1+k^4*x+x*O(x^n))),n)}

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]

A065747 Triangle of Gandhi polynomial coefficients.

Original entry on oeis.org

1, 1, 3, 3, 7, 30, 51, 42, 15, 145, 753, 1656, 1995, 1410, 567, 105, 6631, 39048, 100704, 149394, 140475, 86562, 34566, 8316, 945, 566641, 3656439, 10546413, 17972598, 20133921, 15581349, 8493555, 3246642, 841239, 135135, 10395

Offset: 1

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

First column is A064624.

			Triangle starts
1;
1, 3, 3;
7, 30, 51, 42, 15;
145, 753, 1656, 1995, 1410, 567, 105;
6631 ...

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. A036970, A064624, A065748

Let B(X, n) = X^3 (B(X+1, n-1) - B(X, n-1)), B(X, 1) = X^3; then the (i, j)-th entry is the table is the coefficient of X^(2+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.

A065753 Second column of A065747.

Original entry on oeis.org

3, 30, 753, 39048, 3656439, 562400370, 132584434941, 45454204428636, 21747608978702595, 14048768969900622150, 11925183926773031604489, 13003120724393838963734064, 17866681950868968379286259471

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. A065747, A064624.

cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A065747 Triangle of Gandhi polynomial coefficients.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A065753 Second column of A065747.

Views

Author

Keywords

Links

Crossrefs