A203799 - OEIS

%I #11 Dec 17 2024 15:21:57

%S 1,1,1,8,48,312,2280,18720,171360,1733760,19232640,232243200,

%T 3033676800,42631142400,641383142400,10287038361600,175228365312000,

%U 3159341273088000,60111175348224000,1203646256676864000,25302180885037056000,557134559872450560000,12823826485099069440000

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

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

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

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

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

%e where

%e 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 +...

%o (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)}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) {a(n)=if(n==0||n==1, 1, (n^2 + n - 4)/2 * (n-1)!)}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) {a(n)=n!*polcoeff(1/2 + 1/(2*(1-x)^2) + 2*x + 2*log(1-x +x*O(x^n)), n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A202365, A187742.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Jan 09 2013