0,3

The g.f. is a special case (q=2) of the following identity.

Let W(x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n! = LambertW(-x)/(-x), then

Sum_{n>=0} (n*q^n+1)^(n-1)/q^(n^2)*x^n/n! = Sum_{n>=0} W(x/q^n)^n/q^(n^2)*x^n/n!

where the radius of convergence is |x| <= q/e for q>=1.

G.f.: A(x) = Sum_{n>=0} (n*2^n + 1)^(n-1)/2^(n^2) * x^n/n!

G.f.: A(x) = Sum_{n>=0} W(x/2^n)^n/2^(n^2) * x^n/n!, and

a(n)/2^(n^2) is the coefficient of x^n/n! in W(x)^(1/2^n)

where W(x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.

Radius of convergence of series A(x) is |x| <= 2/e.

G.f.: A(x) = 1 + 3^0/2*x + 9^1/2^4*x^2/2! + 25^2/2^9*x^3/3! + 65^3/2^16*x^4/4! + 161^4/2^25*x^5/5! +...

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

where W(x) = LambertW(-x)/(-x) so that W(x) = exp(x*W(x)).

Special values.

A(1/2) = 1.367881486725746399880346284881720747435653310931858829...

A(1/e) = 1.237164211886302867099485584025040050496738919299895839...

A(2/e) = 2.027079144901937613098735287853530386549370956336296669...

A(-2/e)= 0.733788551140988480682883862465033405661534959498406132...

(PARI) a(n)=(n*2^n + 1)^(n-1)

Sequence in context: A015092 A139107 A085530 this_sequence A157597 A128795 A081232

Adjacent sequences: A158878 A158879 A158880 this_sequence A158882 A158883 A158884

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Apr 22 2009