A142249 - OEIS

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

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OFFSET

1,5

LINKS

Table of n, a(n) for n=1..47.

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

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.

EXAMPLE

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].

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 *)

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).

Sequence in context: A180165 A358349 A376479 * A274705 A257243 A097351

Adjacent sequences: A142246 A142247 A142248 * A142250 A142251 A142252