A187746 G.f.: Sum_{n>=0} (2*n+x)^n * x^n / (1 + 2*n*x + x^2)^n.

1, 2, 13, 100, 984, 11712, 163200, 2603520, 46771200, 934133760, 20530298880, 492355584000, 12793813401600, 358063276032000, 10737974299852800, 343513154086502400, 11676590580695040000, 420271561157640192000, 15967576932074127360000

Offset

0,2

Links

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

Formula

a(n) = (2*n^2+2*n+1) * 2^(n-2) * (n-1)! for n>1 with a(0)=1, a(1)=2.

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

E.g.f.: Sum_{n>=0} a(n+1)*x^n/n! = 2/(1-2*x)^3 + x/(1-2*x).

Example

G.f.: A(x) = 1 + 2*x + 13*x^2 + 100*x^3 + 984*x^4 + 11712*x^5 +...
where
A(x) = 1 + (2+x)*x/(1+2*x+x^2) + (4+x)^2*x^2/(1+4*x+x^2)^2 + (6+x)^3*x^3/(1+6*x+x^2)^3 + (8+x)^4*x^4/(1+8*x+x^2)^4 + (10+x)^5*x^5/(1+10*x+x^2)^5 +...

Prog

(PARI) {a(n)=polcoeff( sum(m=0, n, (2*m+x)^m*x^m/(1+2*m*x+x^2 +x*O(x^n))^m), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=if(n==0, 1, if(n==1, 2, (2*n^2+2*n+1)*2^(n-2)*(n-1)!))}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=n!*polcoeff(1/2 + 1/(2*(1-2*x)^2) - x/2 - log(1-2*x +x*O(x^n))/4, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A187742, A187735.

Sequence in context: A365155 A123619 A341954 * A030519 A141116 A330349

Adjacent sequences: A187743 A187744 A187745 * A187747 A187748 A187749

Keyword

nonn

Author

Paul D. Hanna, Jan 03 2013

Status

approved