A143505 - OEIS

%I #13 Oct 27 2018 02:32:31

%S 1,1,1,1,1,4,3,4,1,1,11,14,23,14,11,1,1,26,70,104,139,104,70,26,1,1,

%T 57,307,530,973,947,973,530,307,57,1,1,120,1197,3016,5970,8568,9549,

%U 8568,5970,3016,1197,120,1,1,247,4300,17101,37105,70474,90069,107241,90069

%N Triangle of coefficients of the polynomials x^(n - 1)*A(n,x + 1/x), where A(n,x) are the Eulerian polynomials of A008292.

%C Row sums yield A000670 (without leading 1).

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

%F Row n is generated by the polynomial (1 - x - 1/x)^(n + 1)*x^(n - 1)*Li(-n, x + 1/x)/(x + 1/x), where Li(n, z) is the polylogarithm function.

%F E.g.f.: (exp(x*y) - exp((1 + x^2)*y))/(x*exp((1 + x^2)*y) - (1 + x^2)*exp(x*y)). - _Franck Maminirina Ramaharo_, Oct 25 2018

%e Triangle begins:

%e    1;

%e    1,  1,   1;

%e    1,  4,   3,   4,   1;

%e    1, 11,  14,  23,  14,  11,   1;

%e    1, 26,  70, 104, 139, 104,  70,  26,   1;

%e    1, 57, 307, 530, 973, 947, 973, 530, 307, 57, 1;

%e     ... reformatted. - _Franck Maminirina Ramaharo_, Oct 25 2018

%t Table[CoefficientList[FullSimplify[ExpandAll[(1 - x - 1/x)^(n + 1)*x^(n - 1)*PolyLog[-n, x + 1/x]/(x + 1/x)]], x], {n, 1, 10}]//Flatten

%Y Compare with A141720.

%Y Cf. A008292.

%Y Cf. A143506, A143507.

%K nonn,tabf

%O 1,6

%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 25 2008

%E Edited and new name by _Franck Maminirina Ramaharo_, Oct 25 2018