A065755
Triangle of Gandhi polynomial coefficients.
Original entry on oeis.org
1, 1, 5, 10, 10, 5, 31, 230, 755, 1440, 1760, 1430, 770, 260, 45, 6721, 60655, 250665, 628535, 1067865, 1299570, 1166945, 783720, 393855, 146025, 38500, 6630, 585, 5850271, 59885980, 285597890, 843288660, 1727996845, 2610132070, 3012643620
Offset: 1
Mike Domaratzki (mdomaratzki(AT)alumni.uwaterloo.ca), Nov 17 2001
Irregular triangle begins:
1;
1, 5, 10, 10, 5;
31, 230, 755, 1440, 1760, 1430, 770, 260, 45;
6721, ...
-
B[X_, 1] := X^5; B[X_, n_] := B[X, n] = X^5 (B[X+1, n-1] - B[X, n-1]) // Expand; row[1] = {1}; row[n_] := List @@ B[X, n] /. X -> 1; Array[row, 5] // Flatten (* Jean-François Alcover, Jul 08 2017 *)
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)
-
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 *)
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