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

%I A142249 #31 Feb 16 2025 08:33:08
%S A142249 -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,
%T A142249 945,3710,2905,455,7,-1,8,2997,20756,31781,12636,1079,8,-1,9,9294,
%U A142249 105299,278304,218559,49754,2469,9
%N A142249 Triangle read by rows, a generalization of the Eulerian numbers based on Nielsen's generalized polylogarithm (case m = 2).
%H A142249 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NielsenGeneralizedPolylogarithm.html">Nielsen Generalized Polylogarithm</a>.
%F A142249 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.
%e A142249 Triangle starts:
%e A142249 {-1}
%e A142249 {-1, 1}
%e A142249 {-1, 2}
%e A142249 {-1, 3,    3}
%e A142249 {-1, 4,   19,      4}
%e A142249 {-1, 5,   80,     65,      5}
%e A142249 {-1, 6,  286,    566,    181,      6}
%e A142249 {-1, 7,  945,   3710,   2905,    455,     7}
%e A142249 {-1, 8, 2997,  20756,  31781,  12636,  1079,    8}
%e A142249 {-1, 9, 9294, 105299, 278304, 218559, 49754, 2469, 9}
%e A142249 ...
%e A142249 For example with n = 4 we have p(n, x ) = (2-1)! * (1 - x)^n * PolyLog(-n, 2, x)/x
%e A142249   = 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].
%t A142249 npl[n_, m_] := (m-1)! (1 - x)^n PolyLog[-n, m, x]/x;
%t A142249 A142249Row[n_] := CoefficientList[FunctionExpand[npl[n, 2]], x] /. Log[1-x] -> 1;
%t A142249 Table[A142249Row[n], {n, 1, 10}] // Flatten
%t A142249 (* Some older versions of Mathematica might use: *)
%t A142249 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 *)
%Y A142249 Row sums are A081047.
%Y A142249 A008292 (m=1), A142249 (m=2), A293298 (m=3 with an additional first column).
%Y A142249 Cf. A293561 (column 3), A293562 (column 4).
%K A142249 sign,tabl
%O A142249 1,5
%A A142249 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 18 2008
%E A142249 Edited by _Peter Luschny_, Oct 11 2017