0,2

E.g.f.: A(x) = Sum_{n>=0} 2^(n^2) * W(2^n*x)^n * x^n/n! ; also, a(n)/n! = coefficient of x^n in W(x)^(2^n) where W(x) = LambertW(-x)/(-x).

E.g.f: A(x) = 1 + 2x + 24x^2/2! + 968x^3/3! + 128000x^4/4! +...

A(x) = 1 + 2*W(2x)*x + 2^4*W(4x)^2*x^2/2! + 2^9*W(8*x)^3*x^3/3! +...

W(x) = LambertW(-x)/(-x) = 1 + x + 3*x^2/2! + 4^2*x^3/3! +...+ (n+1)^(n-1)*x^n/n! +...

This is a special application of the following identity.

Let F(x) be a formal power series in x such that F(0)=1, then

Sum_{n>=0} m^n * F(q^n*x)^b * log( F(q^n*x) )^n / n! =

Sum_{n>=0} x^n * [y^n] F(y)^(m*q^n + b) .

The e.g.f. of this sequence is derived as follows.

Let F(x) = W(x) = LambertW(-x)/(-x);

since log( W(x) ) = x*W(x) and

since W(x)^m = Sum_{n>=0} m*(m+n)^(n-1)*x^n/n! then

Sum_{n>=0} m^n * q^(n^2) * W(q^n*x)^(b+n) * x^n/ n! =

Sum_{n>=0} (m*q^n + b) * (m*q^n + b + n)^(n-1) * x^n.

(PARI) {a(n)=local(W=sum(k=0, n, (k+1)^(k-1)*x^k/k!)); n!*polcoeff( (W+x*O(x^n))^(2^n), n)}

Cf. A136525.

Sequence in context: A012228 A062029 A122551 this_sequence A137887 A094050 A028365

Adjacent sequences: A136521 A136522 A136523 this_sequence A136525 A136526 A136527

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Jan 03 2008