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A152552 Coefficients in a q-analog of the function [LambertW(-2x)/(-2x)]^(1/2) at q=2.

%I A152552 #4 Jul 10 2015 19:37:16
%S A152552 1,1,7,148,7611,872341,213651052,109327540680,115381584785027,
%T A152552 249159124679346991,1095244903267253760231,9765839519517673327876328,
%U A152552 176188639876138769279299798900,6419535615261099235478072782943388
%N A152552 Coefficients in a q-analog of the function [LambertW(-2x)/(-2x)]^(1/2) at q=2.
%F A152552 G.f. satisfies: A(x) = e_q( x*A(x)^2, 2) and A( x/e_q(x,2)^2 ) = e_q(x,2) where e_q(x,q) = Sum_{n>=0} x^n/faq(n,q) is the q-exponential function.
%F A152552 G.f.: A(x) = Sum_{n>=0} a(n)*x^n/faq(n,2) where faq(n,2) = q-factorial of n at q=2.
%F A152552 G.f.: A(x) = [(1/x)*Series_Reversion( x/e_q(x,2)^2 )]^(1/2)
%F A152552 a(n) = Sum_{k=0..n(n-1)/2} A152550(n,k)*2^k.
%e A152552 G.f.: A(x) = 1 + x + 7/3*x^2 + 148/21*x^3 + 7611/315*x^4 + 872341/9765*x^5 +...
%e A152552 e_q(x,2) = 1 + x + x^2/3 + x^3/21 + x^4/315 + x^5/9765 + x^6/615195 +...
%e A152552 The q-factorial of n is faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1).
%o A152552 (PARI) {a(n,q=2)=local(e_q=1+sum(j=1,n,x^j/prod(i=1,j,(q^i-1)/(q-1))), LW2_q=sqrt(serreverse(x/(e_q+x*O(x^n))^2)/x)); polcoeff(LW2_q+x*O(x^n),n,x)*prod(i=1,n,(q^i-1)/(q-1))}
%Y A152552 Cf. A152550, A152551 (q=-1), A152553 (q=3); A005329.
%K A152552 nonn
%O A152552 0,3
%A A152552 _Paul D. Hanna_, Dec 07 2008