A293298 Triangle read by rows, a generalization of the Eulerian numbers based on Nielsen's generalized polylogarithm (case m = 3).

1, 0, 1, 0, 1, -2, 0, 1, -5, 2, 0, 1, -10, 5, 0, 1, -19, 1, 11, 0, 1, -36, -46, 84, 19, 0, 1, -69, -272, 358, 393, 29, 0, 1, -134, -1149, 916, 4171, 1322, 41, 0, 1, -263, -4237, -191, 31939, 26255, 3841, 55, 0, 1, -520, -14536, -20192, 200252, 348848, 130924, 10280, 71

Offset: 0

Author

Peter Luschny, Oct 11 2017

Keywords

• sign
• tabl

Comments

Based on A142249 by Roger L. Bagula and Gary W. Adamson.

Examples

			Triangle starts:
{1}
{0, 1}
{0, 1,   -2}
{0, 1,   -5,     2}
{0, 1,  -10,     5}
{0, 1,  -19,     1,   11}
{0, 1,  -36,   -46,   84,    19}
{0, 1,  -69,  -272,  358,   393,    29}
{0, 1, -134, -1149,  916,  4171,  1322,   41}
{0, 1, -263, -4237, -191, 31939, 26255, 3841, 55}
		

Links

• Eric Weisstein's World of Mathematics, Nielsen Generalized Polylogarithm.

Crossrefs

A123125 (m=1), A142249 (m=2 with missing first column), this seq. (m=3).

Programs

Mathematica

npl[n_, m_] := (m-1)! (1 - x)^n PolyLog[-n, m, x];
A293298Row[0] := {1};
A293298Row[n_] := CoefficientList[FunctionExpand[npl[n, 3]], x] /. Log[1-x] -> 1;
Table[A293298Row[n], {n, 0, 10}] // Flatten

Formula

Let p(n, m) = (m - 1)!*(1 - x)^n*PolyLog(-n, m, x) and P(n) the polynomial given by the expansion of p(n, m=3) after replacing log(1 - x) by 1. T(n, k) is the k-th coefficient of P(n).