A136457 Triangle read by rows: coefficients of polynomials defined by recursion p(x,n)=(x-Gamma(n))*p(x,n-1).
1, -1, 1, 1, -2, 1, -2, 5, -4, 1, 12, -32, 29, -10, 1, -288, 780, -728, 269, -34, 1, 34560, -93888, 88140, -33008, 4349, -154, 1, -24883200, 67633920, -63554688, 23853900, -3164288, 115229, -874, 1, 125411328000, -340899840000, 320383261440, -120287210688, 15971865420, -583918448, 4520189
Offset: 1
Examples
Triangle begins:
{1},
{-1, 1},
{1, -2, 1},
{-2, 5, -4, 1},
{12, -32, 29, -10, 1},
{-288, 780, -728, 269, -34, 1},
{34560, -93888, 88140, -33008, 4349, -154, 1}
Links
- E. F. Cornelius and Phill Schultz, Sequences Generated by Polynomials, Amer. Math. Monthly, No. 2, 2008.
Programs
-
Mathematica
Clear[p, x, n, a] p[x, 0] = 1; p[x, 1] = x - 1; p[x_, m_] := p[x, n] = (x - Gamma[n])*p[x, n - 1]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[p[x, n], x]], {n, 0, 10}]; Table[ExpandAll[p[x, n]], {n, 0, 10}];
Formula
p(x,0)=1; p(x,1)=x-1; p(x,n)=(x-Gamma(n))*p(x,n-1)
Extensions
Edited by N. J. A. Sloane, Aug 10 2008
Comments