A152558 Coefficients in a q-analog of the function LambertW(-2x)/(-2x) at q=3.

1, 2, 22, 912, 126692, 56277344, 78192313656, 335781903409152, 4424572027813470736, 178044609358672673825280, 21805611052892733414074516064, 8108006645142880473904973170212864

Offset

0,2

Links

Table of n, a(n) for n=0..11.

Formula

G.f. satisfies: A(x) = e_q( x*A(x), 3)^2 and A( x/e_q(x,3)^2 ) = e_q(x,3)^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,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 ).

a(n) = Sum_{k=0..n(n-1)/2} A152555(n,k)*3^k.

Example

G.f.: A(x) = 1 + 2*x + 22/4*x^2 + 912/52*x^3 + 126692/2080*x^4 + 56277344/251680*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).

Prog

(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, 3)}

Crossrefs

Cf. A152555, A152556(q=-1), A152557 (q=2) A152559.

Sequence in context: A132568 A279802 A015210 * A202947 A177410 A193486

Adjacent sequences: A152555 A152556 A152557 * A152559 A152560 A152561

Keyword

nonn

Author

Paul D. Hanna, Dec 07 2008

Status

approved