A143505 Triangle of coefficients of the polynomials x^(n - 1)*A(n,x + 1/x), where A(n,x) are the Eulerian polynomials of A008292.

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, 57, 307, 530, 973, 947, 973, 530, 307, 57, 1, 1, 120, 1197, 3016, 5970, 8568, 9549, 8568, 5970, 3016, 1197, 120, 1, 1, 247, 4300, 17101, 37105, 70474, 90069, 107241, 90069

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, Oct 25 2008

Keywords

• nonn
• tabf

Comments

Row sums yield A000670 (without leading 1).

Examples

			Triangle begins:
   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, 57, 307, 530, 973, 947, 973, 530, 307, 57, 1;
    ... reformatted. - _Franck Maminirina Ramaharo_, Oct 25 2018
		

Links

• Eric Weisstein's World of Mathematics, Polylogarithm

Crossrefs

Compare with A141720.

Cf. A008292.

Cf. A143506, A143507.

Programs

Mathematica

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

Formula

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.

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

Extensions

Edited and new name by Franck Maminirina Ramaharo, Oct 25 2018