A152553 Coefficients in a q-analog of the function [LambertW(-2x)/(-2x)]^(1/2) at q=3.
1, 1, 9, 339, 44521, 19059921, 25799597265, 108657870607875, 1410396873934264497, 56078100848527445045121, 6801233273726638573734096441, 2508450630100541880792088526933139
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 9/4*x^2 + 339/52*x^3 + 44521/2080*x^4 + 19059921/251680*x^5 +...
e_q(x,3) = 1 + x + x^2/4 + x^3/52 + x^4/2080 + x^5/251680 + x^6/91611520 +...
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=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))}
Formula
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.
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.
G.f.: A(x) = [(1/x)*Series_Reversion( x/e_q(x,3)^2 )]^(1/2)
a(n) = Sum_{k=0..n(n-1)/2} A152550(n,k)*3^k.