A229039 G.f.: Sum_{n>=0} (n+2)^n * x^n / (1 + (n+2)*x)^n.

1, 3, 7, 24, 108, 600, 3960, 30240, 262080, 2540160, 27216000, 319334400, 4071513600, 56043187200, 828193766400, 13076743680000, 219689293824000, 3912561709056000, 73627297615872000, 1459741204905984000, 30411275102208000000, 664182248232222720000

Offset: 0

Author

Paul D. Hanna, Sep 11 2013

Keywords

• nonn

Comments

More generally, we have the identity:

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.

Examples

			O.g.f.: A(x) = 1 + 3*x + 7*x^2 + 24*x^3 + 108*x^4 + 600*x^5 + 3960*x^6 +...
where
A(x) = 1 + 3*x/(1+3*x) + 4^2*x^2/(1+4*x)^2 + 5^3*x^3/(1+5*x)^3 + 6^4*x^4/(1+6*x)^4 + 7^5*x^5/(1+7*x)^5 +...
E.g.f.: E(x) = 1 + 3*x + 7*x^2/2! + 24*x^3/3! + 108*x^4/4! + 600*x^5/5! +...
where
E(x) = 1 + 3*x + 7/2*x^2 + 4*x^3 + 9/2*x^4 + 5*x^5 + 11/2*x^6 + 6*x^7 +...
which is the expansion of: (2 + 2*x - 3*x^2) / (2 - 4*x + 2*x^2).
		

Crossrefs

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

Programs

Mathematica

a[n_] := (n + 5)*n!/2; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Dec 11 2022 *)

PARI

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

PARI

{a(n)=if(n==0,1, (n+5) * n!/2 )}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) = (n+5) * n!/2 for n>0 with a(0)=1.

E.g.f.: (2 + 2*x - 3*x^2)/(2*(1-x)^2).

From Amiram Eldar, Dec 11 2022: (Start)

Sum_{n>=0} 1/a(n) = 18*e - 237/5.

Sum_{n>=0} (-1)^n/a(n) = 243/5 - 130/e. (End)