A152557 - OEIS

%I #4 Jul 10 2015 19:36:51

%S 1,2,17,394,21377,2537724,637139804,332102399042,355527029604321,

%T 776504491956507890,3445063827264105259985,30955227335240072514768936,

%U 562107762991597803740407081852,20594660519301092842327319372549024

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

%F 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 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 G.f.: A(x) = (1/x)*Series_Reversion( x/e_q(x,2)^2 )

%F a(n) = Sum_{k=0..n(n-1)/2} A152555(n,k)*2^k.

%e G.f.: A(x) = 1 + 2*x + 17/3*x^2 + 394/21*x^3 + 21377/315*x^4 + 2537724/9765*x^5 +...

%e e_q(x,2) = 1 + x + x^2/3 + x^3/21 + x^4/315 + x^5/9765 + x^6/615195 +...

%e The q-factorial of n is faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1).

%o (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 Cf. A152555, A152556(q=-1), A152558 (q=3) A152559.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 07 2008