A187735 G.f.: Sum_{n>=0} (2*n+1)^n * x^n / (1 + (2*n+1)*x)^n.

1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600, 1021870080, 22295347200, 531372441600, 13733933875200, 382588157952000, 11426632984166400, 364223926370304000, 12340763622899712000, 442896294466289664000, 16783438527143608320000

Offset

0,2

Comments

Compare g.f. to the identity (cf. A001710):

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

More generally,

if Sum_{n>=0} a(n)*x^n = Sum_{n>=0} (b*n+c)^n * x^n / (1 + (b*n+c)*x)^n,

then Sum_{n>=0} a(n)*x^n/n! = (2 - 2*(b-c)*x + b*(b-2*c)*x^2)/(2*(1-b*x)^2)

so that a(n) = (b*n + (b+2*c)) * b^(n-1) * n!/2 for n>0 with a(0)=1.

Links

Table of n, a(n) for n=0..18.

Formula

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

E.g.f.: (1-x)/(1-2*x)^2.

From Amiram Eldar, Dec 23 2022: (Start)

Sum_{n>=0} 1/a(n) = 8 - 4*sqrt(e).

Sum_{n>=0} (-1)^n/a(n) = 8 - 12/sqrt(e). (End)

Example

G.f.: A(x) = 1 + 3*x + 16*x^2 + 120*x^3 + 1152*x^4 + 13440*x^5 +...
where
A(x) = 1 + 3*x/(1+3*x) + 5^2*x^2/(1+5*x)^2 + 7^3*x^3/(1+7*x)^3 + 9^4*x^4/(1+9*x)^4 + 11^5*x^5/(1+11*x)^5 +...

Mathematica

a[n_] := (n + 2)*2^(n - 1)*n!; Array[a, 20, 0] (* Amiram Eldar, Dec 23 2022 *)

Prog

(PARI) {a(n)=polcoeff( sum(m=0, n, ((2*m+1)*x)^m / (1 + (2*m+1)*x +x*O(x^n))^m), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = (n+2)*2^(n-1)*n!}

Crossrefs

Cf. A001710, A014479, A187738, A187739, A221160, A221161, A187740.

Sequence in context: A199670 A192783 A136168 * A200318 A120015 A003692

Adjacent sequences: A187732 A187733 A187734 * A187736 A187737 A187738

Keyword

nonn

Author

Paul D. Hanna, Jan 02 2013

Status

approved