A152557 Coefficients in a q-analog of the function LambertW(-2x)/(-2x) at q=2.

1, 2, 17, 394, 21377, 2537724, 637139804, 332102399042, 355527029604321, 776504491956507890, 3445063827264105259985, 30955227335240072514768936, 562107762991597803740407081852, 20594660519301092842327319372549024

Offset

0,2

Links

Table of n, a(n) for n=0..13.

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)^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 )

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

Example

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

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, 2)}

Crossrefs

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

Sequence in context: A198405 A202972 A085617 * A015202 A308864 A370704

Adjacent sequences: A152554 A152555 A152556 * A152558 A152559 A152560

Keyword

nonn

Author

Paul D. Hanna, Dec 07 2008

Status

approved