A136525 a(n) = (2^n + 1)*(2^n + n + 1)^(n-1).

1, 3, 35, 1296, 157437, 68809488, 117274907815, 816249936543744, 23585997104539765625, 2828012919296320973299968, 1396969787088550953695654296875, 2819773093146732354646026240000000000

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..57

Formula

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

a(n)/n! = coefficient of x^n in W(x)^(2^n+1) where W(x) = LambertW(-x)/(-x).

Example

E.g.f: A(x) = 1 + 3x + 35x^2/2! + 1296x^3/3! + 157437x^4/4! +...
A(x) = W(x) + 2*W(2x)^2*x + 2^4*W(4x)^3*x^2/2! + 2^9*W(8*x)^4*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.

Mathematica

Table[(2^n +1)*(2^n +n+1)^(n-1), {n, 0, 30}] (* G. C. Greubel, Sep 20 2022 *)

Prog

(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+1), n)
(PARI) /* As coefficient of x^n in Series: */ a(n)=local(W=sum(k=0, n, (k+1)^(k-1)*x^k/k!)); n!*polcoeff(sum(i=0, n, 2^(i^2)*subst(W, x, 2^i*x+x*O(x^n))^(i+1)*x^i/i!), n)
(Magma) [(2^n +1)*(2^n +n+1)^(n-1): n in [0..30]]; // G. C. Greubel, Sep 20 2022
(SageMath) [(2^n +1)*(2^n +n+1)^(n-1) for n in (0..30)] # G. C. Greubel, Sep 20 2022

Crossrefs

Cf. A136524.

Sequence in context: A012767 A279377 A215582 * A136556 A006098 A320845

Adjacent sequences: A136522 A136523 A136524 * A136526 A136527 A136528

Keyword

nonn

Author

Paul D. Hanna, Jan 03 2008

Status

approved