A142249 Triangle read by rows, a generalization of the Eulerian numbers based on Nielsen's generalized polylogarithm (case m = 2).

-1, -1, 1, -1, 2, -1, 3, 3, -1, 4, 19, 4, -1, 5, 80, 65, 5, -1, 6, 286, 566, 181, 6, -1, 7, 945, 3710, 2905, 455, 7, -1, 8, 2997, 20756, 31781, 12636, 1079, 8, -1, 9, 9294, 105299, 278304, 218559, 49754, 2469, 9

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, Sep 18 2008

Keywords

• sign
• tabl

Examples

			Triangle starts:
{-1}
{-1, 1}
{-1, 2}
{-1, 3,    3}
{-1, 4,   19,      4}
{-1, 5,   80,     65,      5}
{-1, 6,  286,    566,    181,      6}
{-1, 7,  945,   3710,   2905,    455,     7}
{-1, 8, 2997,  20756,  31781,  12636,  1079,    8}
{-1, 9, 9294, 105299, 278304, 218559, 49754, 2469, 9}
...
For example with n = 4 we have p(n, x ) = (2-1)! * (1 - x)^n * PolyLog(-n, 2, x)/x
  = x*(7 + 4*x) - (1 + 4*x + x^2)*log(1-x). Replacing log(1-x) by 1 reduces this to x*(7 + 4*x) - (1 + 4*x + x^2) = 3*x^2 + 3*x - 1 with coefficients [-1, 3, 3].
		

Links

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

Crossrefs

Row sums are A081047.

A008292 (m=1), A142249 (m=2), A293298 (m=3 with an additional first column).

Cf. A293561 (column 3), A293562 (column 4).

Programs

Mathematica

npl[n_, m_] := (m-1)! (1 - x)^n PolyLog[-n, m, x]/x;
A142249Row[n_] := CoefficientList[FunctionExpand[npl[n, 2]], x] /. Log[1-x] -> 1;
Table[A142249Row[n], {n, 1, 10}] // Flatten
(* Some older versions of Mathematica might use: *)
Flatten[Table[CoefficientList[Simplify[(1-x)^n * PolyLog[-n, 2, x] / (x*Log[1-x])], x]/.x->1-E, {n, 1, 15}]] (* Vaclav Kotesovec, Oct 12 2017 *)

Formula

Let p(n, m) = (m - 1)!*(1 - x)^n*PolyLog(-n, m, x)/x and P(n) the polynomial given by the expansion of p(n, m=2) after replacing log(1 - x) by 1. T(n, k) is the k-th coefficient of P(n). Using instead p(n, m=1) gives the Eulerian numbers A008292.

Extensions

Edited by Peter Luschny, Oct 11 2017