A202365 - OEIS

%I #19 Dec 21 2024 20:06:39

%S 1,1,2,10,54,336,2400,19440,176400,1774080,19595520,235872000,

%T 3073593600,43110144000,647610163200,10374216652800,176536039680000,

%U 3180264062976000,60466862776320000,1210048630382592000,25423825985445888000,559567461880627200000,12874917427270778880000

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

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

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

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

%F From _Amiram Eldar_, Dec 23 2022: (Start)

%F Sum_{n>=0} 1/a(n) = Pi^2/9 + 43/27.

%F Sum_{n>=0} (-1)^n/a(n) = Pi^2/18 - 4*log(2)/9 + 5/27. (End)

%e G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 54*x^4 + 336*x^5 + 2400*x^6 +...

%e where

%e A(x) = 1 + (1-x)*x/(1+x-x^2) + (2-x)^2*x^2/(1+2*x-x^2)^2 + (3-x)^3*x^3/(1+3*x-x^2)^3 + (4-x)^4*x^4/(1+4*x-x^2)^4 + (5-x)^5*x^5/(1+5*x-x^2)^5 +...

%t a[n_] := Switch[n, 0|1, 1, _, (n-1)*(n+2)/2*(n-1)!];

%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Aug 24 2022 *)

%o (PARI) {a(n)=polcoeff( sum(m=0, n, (m-x)^m*x^m/(1+m*x-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-1)*(n+2)/2 * (n-1)!)}

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

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

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

%Y Cf. A187742, A187741, A187735, A187746.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 09 2013