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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.

%I A143506 #17 Oct 27 2018 02:52:20
%S A143506 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,
%T A143506 237,1687,2630,5293,4787,5293,2630,1687,237,1,1,722,10549,27158,52730,
%U A143506 78586,84365,78586,52730,27158,10549,722,1,1,2179,60664,272797,563029,1132234
%N 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.
%C A143506 Row sums yield A080253.
%H A143506 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lerch_zeta_function">Lerch zeta function</a>
%F A143506 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.
%F A143506 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
%e A143506 Triangle begins:
%e A143506    1;
%e A143506    1,   1,    1;
%e A143506    1,   6,    3,    6,    1;
%e A143506    1,  23,   26,   47,   26,   23,    1;
%e A143506    1,  76,  234,  304,  467,  304,  234,   76,    1;
%e A143506    1, 237, 1687, 2630, 5293, 4787, 5293, 2630, 1687, 237, 1;
%e A143506     ... reformatted. - _Franck Maminirina Ramaharo_, Oct 25 2018
%t A143506 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
%Y A143506 Cf. A008292, A060187.
%Y A143506 Cf. A143505, A143507.
%K A143506 nonn,tabf
%O A143506 0,6
%A A143506 _Roger L. Bagula_ and _Gary W. Adamson_, Oct 25 2008
%E A143506 Edited, new name, and offset corrected by _Franck Maminirina Ramaharo_, Oct 25 2018