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A152557 Coefficients in a q-analog of the function LambertW(-2x)/(-2x) at q=2.

%I A152557 #4 Jul 10 2015 19:36:51
%S A152557 1,2,17,394,21377,2537724,637139804,332102399042,355527029604321,
%T A152557 776504491956507890,3445063827264105259985,30955227335240072514768936,
%U A152557 562107762991597803740407081852,20594660519301092842327319372549024
%N A152557 Coefficients in a q-analog of the function LambertW(-2x)/(-2x) at q=2.
%F A152557 G.f. satisfies: A(x) = e_q( x*A(x), 2)^2 and A( x/e_q(x,2)^2 ) = e_q(x,2)^2 where e_q(x,q) = Sum_{n>=0} x^n/faq(n,q) is the q-exponential function.
%F A152557 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 A152557 G.f.: A(x) = (1/x)*Series_Reversion( x/e_q(x,2)^2 )
%F A152557 a(n) = Sum_{k=0..n(n-1)/2} A152555(n,k)*2^k.
%e A152557 G.f.: A(x) = 1 + 2*x + 17/3*x^2 + 394/21*x^3 + 21377/315*x^4 + 2537724/9765*x^5 +...
%e A152557 e_q(x,2) = 1 + x + x^2/3 + x^3/21 + x^4/315 + x^5/9765 + x^6/615195 +...
%e A152557 The q-factorial of n is faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1).
%o A152557 (PARI) {a(n)=local(e_q=1+sum(j=1,n,x^j/prod(i=1,j,(q^i-1)/(q-1))), LW2_q=serreverse(x/(e_q+x*O(x^n))^2)/x); subst(polcoeff(LW2_q+x*O(x^n),n,x)*prod(i=1,n,(q^i-1)/(q-1)),q,2)}
%Y A152557 Cf. A152555, A152556(q=-1), A152558 (q=3) A152559.
%K A152557 nonn
%O A152557 0,2
%A A152557 _Paul D. Hanna_, Dec 07 2008