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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. 8

%I #14 Oct 22 2018 09:57:42

%S 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 -532,-24,904,-672,48,80,-16,1,113,980,-1094,-5128,8760,-736,-6224,

%U 3920,-432,-192,32,1,239,4284,5276,-43964,19764,90272,-114080,19824,36304

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

%F 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 Triangle begins:

%e 1;

%e 1, -1;

%e 1, 1, -4, 2;

%e 1, 7, -12, -4, 12, -4;

%e 1, 21, 0, -102, 100, 4, -32, 8;

%e 1, 51, 160, -532, -24, 904, -672, 48, 80, -16;

%e ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018

%t p[x_, n_] = If[n == 0, 1, (1 + 2*(-1 + x)*x)^(n + 1)*PolyLog[-n, 2*x*(1 - x)]/(2*x)];

%t Table[CoefficientList[FullSimplify[Expand[p[x, n]]], x], {n, 0, 10}]//Flatten

%Y Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.

%Y Cf. A142175, A168287, A168288, A168289, A168290, A168291, A168292, A168293.

%K sign,tabf

%O 0,6

%A _Roger L. Bagula_ and _Gary W. Adamson_, Sep 15 2008

%E Edited, new name, and offset corrected by _Franck Maminirina Ramaharo_, Oct 21 2018

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)