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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.
2
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
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.
Sequence in context: A070511 A066340 A195597 * A245727 A280822 A346785
KEYWORD
nonn,tabf
AUTHOR
EXTENSIONS
Edited and new name by Franck Maminirina Ramaharo, Oct 25 2018
STATUS
approved