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

%I A293298 #15 Feb 16 2025 08:33:51
%S A293298 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,
%T A293298 -69,-272,358,393,29,0,1,-134,-1149,916,4171,1322,41,0,1,-263,-4237,
%U A293298 -191,31939,26255,3841,55,0,1,-520,-14536,-20192,200252,348848,130924,10280,71
%N A293298 Triangle read by rows, a generalization of the Eulerian numbers based on Nielsen's generalized polylogarithm (case m = 3).
%C A293298 Based on A142249 by _Roger L. Bagula_ and _Gary W. Adamson_.
%H A293298 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NielsenGeneralizedPolylogarithm.html">Nielsen Generalized Polylogarithm</a>.
%F A293298 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).
%e A293298 Triangle starts:
%e A293298 {1}
%e A293298 {0, 1}
%e A293298 {0, 1,   -2}
%e A293298 {0, 1,   -5,     2}
%e A293298 {0, 1,  -10,     5}
%e A293298 {0, 1,  -19,     1,   11}
%e A293298 {0, 1,  -36,   -46,   84,    19}
%e A293298 {0, 1,  -69,  -272,  358,   393,    29}
%e A293298 {0, 1, -134, -1149,  916,  4171,  1322,   41}
%e A293298 {0, 1, -263, -4237, -191, 31939, 26255, 3841, 55}
%t A293298 npl[n_, m_] := (m-1)! (1 - x)^n PolyLog[-n, m, x];
%t A293298 A293298Row[0] := {1};
%t A293298 A293298Row[n_] := CoefficientList[FunctionExpand[npl[n, 3]], x] /. Log[1-x] -> 1;
%t A293298 Table[A293298Row[n], {n, 0, 10}] // Flatten
%Y A293298 A123125 (m=1), A142249 (m=2 with missing first column), this seq. (m=3).
%K A293298 sign,tabl
%O A293298 0,6
%A A293298 _Peter Luschny_, Oct 11 2017