A136524 a(n) = 2^n*(2^n + n)^(n-1).

1, 2, 24, 968, 128000, 59973152, 107564800000, 774840978000000, 22880602349081985024, 2779532441951756299960832, 1383517973370847653192530395136, 2804745232514434754535719279455029248, 23030042114303275003004737650852694413279232

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 * x^n/n!.

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

Example

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

Mathematica

Table[2^n*(2^n +n)^(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), n)}
(Magma) [2^n*(2^n + n)^(n-1): n in [0..30]]; // G. C. Greubel, Sep 20 2022
(SageMath) [2^n*(2^n + n)^(n-1) for n in (0..30)] # G. C. Greubel, Sep 20 2022

Crossrefs

Cf. A136525.

Sequence in context: A012228 A062029 A122551 * A213984 A268311 A362091

Adjacent sequences: A136521 A136522 A136523 * A136525 A136526 A136527

Keyword

nonn,easy

Author

Paul D. Hanna, Jan 03 2008

Status

approved