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

0,6

Links

Table of n, a(n) for n=0..52.

Eric Weisstein's World of Mathematics, Polylogarithm

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

Example

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

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

Crossrefs

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

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

Sequence in context: A039962 A046741 A136249 * A291977 A142073 A193559

Adjacent sequences: A142144 A142145 A142146 * A142148 A142149 A142150

Keyword

sign,tabf

Author

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

Extensions

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

Status

approved