A152283 Coefficients in a q-analog of the LambertW function at q=3: A(x) = Sum_{n>=0} a(n)*x^n/faq(n,3) where faq(n,q) = q-factorial of n.
1, 1, 5, 92, 5621, 1093236, 663362421, 1242109529088, 7129029760138649, 124860091946887218320, 6652206059042029394600021, 1075572123264132205051327968256, 526826946994724781414669857330392909
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 5/4*x^2 + 92/52*x^3 + 5621/2080*x^4 + 1093236/251680*x^5 +...
G.f. satisfies: A(x) = e_q( x*A(x), 3) where the q-exponential series is:
e_q(x,q) = 1 + x + x^2/faq(2,q) + x^3/faq(3,q) +...+ x^n/faq(n,q) +...
e_q(x,3) = 1 + x + x^2/4 + x^3/52 + x^4/2080 + x^5/251680 +...
The q-factorial of n is faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1):
faq(0,q)=1, faq(1,q)=1, faq(2,q)=(1+q), faq(3,q)=(1+q)*(1+q+q^2), faq(4,q)=(1+q)*(1+q+q^2)*(1+q+q^2+q^3), ...
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))),LW_q=serreverse(x/e_q+x^2*O(x^n))/x); polcoeff(LW_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), 3) and A( x/e_q(x,3) ) = 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) = (1/x)*Series_Reversion( x/e_q(x,3) ).
a(n) = Sum_{k=0..n(n-1)/2} A152290(n,k)*3^k.
a(n) = faq(n,3)*Sum_{pi} n!/((n-k+1)!*Product_{i=1..n} (e(i)!*faq(i,3)^e(i))), where pi runs through all nonnegative integer solutions of e(1)+2*e(2)+...+n*e(n)=n and k=e(1)+e(2)+...+e(n). [From Vladeta Jovovic, Dec 03 2008]