A143506 Irregular triangle read by rows: first row is 1, and n-th row gives the coefficients of x^(n - 1)*R(n,x + 1/x)/(x + 1/x), where R(n,x) is the n-th row polynomial for A060187.

1, 1, 1, 1, 1, 6, 3, 6, 1, 1, 23, 26, 47, 26, 23, 1, 1, 76, 234, 304, 467, 304, 234, 76, 1, 1, 237, 1687, 2630, 5293, 4787, 5293, 2630, 1687, 237, 1, 1, 722, 10549, 27158, 52730, 78586, 84365, 78586, 52730, 27158, 10549, 722, 1, 1, 2179, 60664, 272797, 563029, 1132234

Offset: 0

Author

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

Keywords

• nonn
• tabf

Comments

Row sums yield A080253.

Examples

			Triangle begins:
   1;
   1,   1,    1;
   1,   6,    3,    6,    1;
   1,  23,   26,   47,   26,   23,    1;
   1,  76,  234,  304,  467,  304,  234,   76,    1;
   1, 237, 1687, 2630, 5293, 4787, 5293, 2630, 1687, 237, 1;
    ... reformatted. - _Franck Maminirina Ramaharo_, Oct 25 2018
		

Links

• Wikipedia, Lerch zeta function

Crossrefs

Cf. A008292, A060187.

Cf. A143505, A143507.

Programs

Mathematica

Table[CoefficientList[FullSimplify[ExpandAll[2^n*(1 - x - 1/x)^(1 + n)*x^n*LerchPhi[x + 1/x, -n, 1/2]]], x], {n, 0, 10}]//Flatten

Formula

Row n is generated by the polynomial 2^n*(1 - x - 1/x)^(1 + n)*x^n*Phi(x + 1/x, -n, 1/2), where Phi is the Lerch transcendant.

E.g.f.: (1 - x + x^2)*exp((1 + x + x^2)*t)/((1 + x^2)*exp(2*t*x) - x*exp(2*(1 + x^2)*t)). - Franck Maminirina Ramaharo, Oct 25 2018

Extensions

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