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

1,6

Comments

Row sums yield A000670 (without leading 1).

Links

Table of n, a(n) for n=1..58.

Eric Weisstein's World of Mathematics, Polylogarithm

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

Example

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

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

Crossrefs

Compare with A141720.

Cf. A008292.

Cf. A143506, A143507.

Sequence in context: A070511 A066340 A195597 * A245727 A280822 A346785

Adjacent sequences: A143502 A143503 A143504 * A143506 A143507 A143508

Keyword

nonn,tabf

Author

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

Extensions

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

Status

approved