A142147 Irregular triangle read by rows: first row is 1, and the n-th row gives the coefficients in the expansion of (1/2*x)*(1 - 2*x*(1 - x))^(n + 1)*Li(-n, 2*x*(1 - x)), where Li(n, z) is the polylogarithm.

1, 1, -1, 1, 1, -4, 2, 1, 7, -12, -4, 12, -4, 1, 21, 0, -102, 100, 4, -32, 8, 1, 51, 160, -532, -24, 904, -672, 48, 80, -16, 1, 113, 980, -1094, -5128, 8760, -736, -6224, 3920, -432, -192, 32, 1, 239, 4284, 5276, -43964, 19764, 90272, -114080, 19824, 36304

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Sep 15 2008

Keywords

• sign
• tabf

Examples

			Triangle begins:
     1;
     1, -1;
     1,  1,  -4,    2;
     1,  7, -12,   -4,  12,  -4;
     1, 21,   0, -102, 100,   4,  -32,  8;
     1, 51, 160, -532, -24, 904, -672, 48, 80, -16;
      ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018
		

Links

• Eric Weisstein's World of Mathematics, Polylogarithm

Crossrefs

Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.

Cf. A142175, A168287, A168288, A168289, A168290, A168291, A168292, A168293.

Programs

Mathematica

p[x_, n_] = If[n == 0, 1, (1 + 2*(-1 + x)*x)^(n + 1)*PolyLog[-n, 2*x*(1 - x)]/(2*x)];
Table[CoefficientList[FullSimplify[Expand[p[x, n]]], x], {n, 0, 10}]//Flatten

Formula

E.g.f.: ((1 - x)*(1 - 2*x)*exp(t*(1 + 2*x^2)) + x*exp(2*t*x))/(exp(2*t*x) - 2*x*(1 - x)*exp(t*(1 + 2*x^2))). - Franck Maminirina Ramaharo, Oct 22 2018

Extensions

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 21 2018