A152552 Coefficients in a q-analog of the function [LambertW(-2x)/(-2x)]^(1/2) at q=2.
1, 1, 7, 148, 7611, 872341, 213651052, 109327540680, 115381584785027, 249159124679346991, 1095244903267253760231, 9765839519517673327876328, 176188639876138769279299798900, 6419535615261099235478072782943388
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 7/3*x^2 + 148/21*x^3 + 7611/315*x^4 + 872341/9765*x^5 +...
e_q(x,2) = 1 + x + x^2/3 + x^3/21 + x^4/315 + x^5/9765 + x^6/615195 +...
The q-factorial of n is faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1).
Programs
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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))}
Formula
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.
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.
G.f.: A(x) = [(1/x)*Series_Reversion( x/e_q(x,2)^2 )]^(1/2)
a(n) = Sum_{k=0..n(n-1)/2} A152550(n,k)*2^k.