A203799 G.f.: Sum_{n>=0} (n-2*x)^n * x^n / (1 + n*x - 2*x^2)^n.

1, 1, 1, 8, 48, 312, 2280, 18720, 171360, 1733760, 19232640, 232243200, 3033676800, 42631142400, 641383142400, 10287038361600, 175228365312000, 3159341273088000, 60111175348224000, 1203646256676864000, 25302180885037056000, 557134559872450560000, 12823826485099069440000

Offset

0,4

Links

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

Formula

a(n) = (n^2 + n - 4)/2 * (n-1)!, for n>1 with a(0)=a(1)=1.

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

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

Example

G.f.: A(x) = 1 + x + x^2 + 8*x^3 + 48*x^4 + 312*x^5 + 2280*x^6 +...
where
A(x) = 1 + (1-2*x)*x/(1+x-2*x^2) + (2-2*x)^2*x^2/(1+2*x-2*x^2)^2 + (3-2*x)^3*x^3/(1+3*x-2*x^2)^3 + (4-2*x)^4*x^4/(1+4*x-2*x^2)^4 + (5-2*x)^5*x^5/(1+5*x-2*x^2)^5 +...

Prog

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

Crossrefs

Cf. A202365, A187742.

Sequence in context: A037507 A037690 A180031 * A095897 A220251 A025013

Adjacent sequences: A203796 A203797 A203798 * A203800 A203801 A203802

Keyword

nonn

Author

Paul D. Hanna, Jan 09 2013

Status

approved