%I #10 Dec 11 2022 06:02:49
%S 1,4,9,30,132,720,4680,35280,302400,2903040,30844800,359251200,
%T 4550515200,62270208000,915372057600,14384418048000,240612083712000,
%U 4268249137152000,80029671321600000,1581386305314816000,32844177110384640000,715273190403932160000,16298010552775311360000
%N G.f.: Sum_{n>=0} (n+3)^n * x^n / (1 + (n+3)*x)^n.
%F a(n) = (n+7) * n!/2 for n>0 with a(0)=1.
%F E.g.f.: (2 + 4*x - 5*x^2)/(2*(1-x)^2).
%F From _Amiram Eldar_, Dec 11 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 530*e - 10075/7.
%F Sum_{n>=0} (-1)^n/a(n) = 10085/7 - 3914/e. (End)
%e O.g.f.: A(x) = 1 + 4*x + 9*x^2 + 30*x^3 + 132*x^4 + 720*x^5 + 4680*x^6 +...
%e where
%e A(x) = 1 + 4*x/(1+4*x) + 5^2*x^2/(1+5*x)^2 + 6^3*x^3/(1+6*x)^3 + 7^4*x^4/(1+7*x)^4 + 8^5*x^5/(1+8*x)^5 +...
%e E.g.f.: E(x) = 1 + 4*x + 9*x^2/2! + 30*x^3/3! + 132*x^4/4! + 720*x^5/5! +...
%e where
%e E(x) = 1 + 4*x + 9/2*x^2 + 5*x^3 + 11/2*x^4 + 6*x^5 + 13/2*x^6 + 7*x^7 +...
%e which is the expansion of: (2 + 4*x - 5*x^2) / (2 - 4*x + 2*x^2).
%p a:=series(add((n+3)^n*x^n/(1+(n+3)*x)^n,n=0..100),x=0,23): seq(coeff(a,x,n),n=0..22); # _Paolo P. Lava_, Mar 27 2019
%t a[n_] := (n + 7)*n!/2; a[0] = 1; Array[a, 25, 0] (* _Amiram Eldar_, Dec 11 2022 *)
%o (PARI) {a(n)=polcoeff( sum(m=0, n, ((m+3)*x)^m / (1 + (m+3)*x +x*O(x^n))^m), n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n)=if(n==0, 1, (n+7) * n!/2 )}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A229039, A038720, A230056, A187735, A187738, A187739, A229039, A221160, A221161, A187740.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Oct 07 2013