A230056 G.f.: Sum_{n>=0} (n+3)^n * x^n / (1 + (n+3)*x)^n.

1, 4, 9, 30, 132, 720, 4680, 35280, 302400, 2903040, 30844800, 359251200, 4550515200, 62270208000, 915372057600, 14384418048000, 240612083712000, 4268249137152000, 80029671321600000, 1581386305314816000, 32844177110384640000, 715273190403932160000, 16298010552775311360000

Offset

0,2

Links

Table of n, a(n) for n=0..22.

Formula

a(n) = (n+7) * n!/2 for n>0 with a(0)=1.

E.g.f.: (2 + 4*x - 5*x^2)/(2*(1-x)^2).

From Amiram Eldar, Dec 11 2022: (Start)

Sum_{n>=0} 1/a(n) = 530*e - 10075/7.

Sum_{n>=0} (-1)^n/a(n) = 10085/7 - 3914/e. (End)

Example

O.g.f.: A(x) = 1 + 4*x + 9*x^2 + 30*x^3 + 132*x^4 + 720*x^5 + 4680*x^6 +...
where
A(x) = 1 + 4*x/(1+4*x) + 5^2*x^2/(1+5*x)^2 + 6^3*x^3/(1+6*x)^3 + 7^4*x^4/(1+7*x)^4 + 8^5*x^5/(1+8*x)^5 +...
E.g.f.: E(x) = 1 + 4*x + 9*x^2/2! + 30*x^3/3! + 132*x^4/4! + 720*x^5/5! +...
where
E(x) = 1 + 4*x + 9/2*x^2 + 5*x^3 + 11/2*x^4 + 6*x^5 + 13/2*x^6 + 7*x^7 +...
which is the expansion of: (2 + 4*x - 5*x^2) / (2 - 4*x + 2*x^2).

Maple

a:=series(add((n+3)^n*x^n/(1+(n+3)*x)^n, n=0..100), x=0, 23): seq(coeff(a, x, n), n=0..22); # Paolo P. Lava, Mar 27 2019

Mathematica

a[n_] := (n + 7)*n!/2; a[0] = 1; Array[a, 25, 0] (* Amiram Eldar, Dec 11 2022 *)

Prog

(PARI) {a(n)=polcoeff( sum(m=0, n, ((m+3)*x)^m / (1 + (m+3)*x +x*O(x^n))^m), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=if(n==0, 1, (n+7) * n!/2 )}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A229039, A038720, A230056, A187735, A187738, A187739, A229039, A221160, A221161, A187740.

Sequence in context: A149114 A105865 A187983 * A069103 A364239 A041137

Adjacent sequences: A230053 A230054 A230055 * A230057 A230058 A230059

Keyword

nonn

Author

Paul D. Hanna, Oct 07 2013

Status

approved