cp's OEIS Frontend

A152553 Coefficients in a q-analog of the function [LambertW(-2x)/(-2x)]^(1/2) at q=3.

%I A152553 #4 Jul 10 2015 19:37:08
%S A152553 1,1,9,339,44521,19059921,25799597265,108657870607875,
%T A152553 1410396873934264497,56078100848527445045121,
%U A152553 6801233273726638573734096441,2508450630100541880792088526933139
%N A152553 Coefficients in a q-analog of the function [LambertW(-2x)/(-2x)]^(1/2) at q=3.
%F A152553 G.f. satisfies: A(x) = e_q( x*A(x)^2, 3) and A( x/e_q(x,3)^2 ) = e_q(x,3) where e_q(x,q) = Sum_{n>=0} x^n/faq(n,q) is the q-exponential function.
%F A152553 G.f.: A(x) = Sum_{n>=0} a(n)*x^n/faq(n,3) where faq(n,3) = q-factorial of n at q=3.
%F A152553 G.f.: A(x) = [(1/x)*Series_Reversion( x/e_q(x,3)^2 )]^(1/2)
%F A152553 a(n) = Sum_{k=0..n(n-1)/2} A152550(n,k)*3^k.
%e A152553 G.f.: A(x) = 1 + x + 9/4*x^2 + 339/52*x^3 + 44521/2080*x^4 + 19059921/251680*x^5 +...
%e A152553 e_q(x,3) = 1 + x + x^2/4 + x^3/52 + x^4/2080 + x^5/251680 + x^6/91611520 +...
%e A152553 The q-factorial of n is faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1).
%o A152553 (PARI) {a(n,q=3)=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 A152553 Cf. A152550, A152551 (q=-1), A152552 (q=2); A015001.
%K A152553 nonn
%O A152553 0,3
%A A152553 _Paul D. Hanna_, Dec 07 2008