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Triangle read by rows, a generalization of the Eulerian numbers based on Nielsen's generalized polylogarithm (case m = 3).
1

%I #14 Oct 11 2017 19:43:17

%S 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 -69,-272,358,393,29,0,1,-134,-1149,916,4171,1322,41,0,1,-263,-4237,

%U -191,31939,26255,3841,55,0,1,-520,-14536,-20192,200252,348848,130924,10280,71

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

%C Based on A142249 by _Roger L. Bagula_ and _Gary W. Adamson_.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NielsenGeneralizedPolylogarithm.html">Nielsen Generalized Polylogarithm</a>.

%F 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 Triangle starts:

%e {1}

%e {0, 1}

%e {0, 1, -2}

%e {0, 1, -5, 2}

%e {0, 1, -10, 5}

%e {0, 1, -19, 1, 11}

%e {0, 1, -36, -46, 84, 19}

%e {0, 1, -69, -272, 358, 393, 29}

%e {0, 1, -134, -1149, 916, 4171, 1322, 41}

%e {0, 1, -263, -4237, -191, 31939, 26255, 3841, 55}

%t npl[n_, m_] := (m-1)! (1 - x)^n PolyLog[-n, m, x];

%t A293298Row[0] := {1};

%t A293298Row[n_] := CoefficientList[FunctionExpand[npl[n, 3]], x] /. Log[1-x] -> 1;

%t Table[A293298Row[n], {n, 0, 10}] // Flatten

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

%K sign,tabl

%O 0,6

%A _Peter Luschny_, Oct 11 2017