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A142147 Irregular triangle read by rows: first row is 1, and the n-th row gives the coefficients in the expansion of (1/2*x)*(1 - 2*x*(1 - x))^(n + 1)*Li(-n, 2*x*(1 - x)), where Li(n, z) is the polylogarithm.

%I A142147 #15 Feb 16 2025 08:33:08
%S A142147 1,1,-1,1,1,-4,2,1,7,-12,-4,12,-4,1,21,0,-102,100,4,-32,8,1,51,160,
%T A142147 -532,-24,904,-672,48,80,-16,1,113,980,-1094,-5128,8760,-736,-6224,
%U A142147 3920,-432,-192,32,1,239,4284,5276,-43964,19764,90272,-114080,19824,36304
%N A142147 Irregular triangle read by rows: first row is 1, and the n-th row gives the coefficients in the expansion of (1/2*x)*(1 - 2*x*(1 - x))^(n + 1)*Li(-n, 2*x*(1 - x)), where Li(n, z) is the polylogarithm.
%H A142147 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>
%F A142147 E.g.f.: ((1 - x)*(1 - 2*x)*exp(t*(1 + 2*x^2)) + x*exp(2*t*x))/(exp(2*t*x) - 2*x*(1 - x)*exp(t*(1 + 2*x^2))). - _Franck Maminirina Ramaharo_, Oct 22 2018
%e A142147 Triangle begins:
%e A142147      1;
%e A142147      1, -1;
%e A142147      1,  1,  -4,    2;
%e A142147      1,  7, -12,   -4,  12,  -4;
%e A142147      1, 21,   0, -102, 100,   4,  -32,  8;
%e A142147      1, 51, 160, -532, -24, 904, -672, 48, 80, -16;
%e A142147       ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018
%t A142147 p[x_, n_] = If[n == 0, 1, (1 + 2*(-1 + x)*x)^(n + 1)*PolyLog[-n, 2*x*(1 - x)]/(2*x)];
%t A142147 Table[CoefficientList[FullSimplify[Expand[p[x, n]]], x], {n, 0, 10}]//Flatten
%Y A142147 Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.
%Y A142147 Cf. A142175, A168287, A168288, A168289, A168290, A168291, A168292, A168293.
%K A142147 sign,tabf
%O A142147 0,6
%A A142147 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 15 2008
%E A142147 Edited, new name, and offset corrected by _Franck Maminirina Ramaharo_, Oct 21 2018