A158881 a(n) = (n*2^n + 1)^(n-1).

1, 1, 9, 625, 274625, 671898241, 8458700490625, 520900360822838529, 151632993506657159886849, 203635581444958952230203985921, 1239028497632876493535705227172341761

Offset

0,3

Comments

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.

Links

G. C. Greubel, Table of n, a(n) for n = 0..55

Formula

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.

Example

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...

Maple

seq( (n*2^n+1)^(n-1), n=0..10); # G. C. Greubel, Mar 04 2020

Mathematica

Table[(n*2^n+1)^(n-1), {n, 0, 10}] (* Harvey P. Dale, Jun 04 2015 *)

Prog

(PARI) a(n)=(n*2^n + 1)^(n-1);
(Magma) [(n*2^n+1)^(n-1): n in [0..10]]; // G. C. Greubel, Mar 04 2020
(Sage) [(n*2^n+1)^(n-1) for n in (0..10)] # G. C. Greubel, Mar 04 2020

Crossrefs

Sequence in context: A352276 A085530 A188423 * A188394 A157597 A211611

Adjacent sequences: A158878 A158879 A158880 * A158882 A158883 A158884

Keyword

nonn

Author

Paul D. Hanna, Apr 22 2009

Status

approved