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A229039 G.f.: Sum_{n>=0} (n+2)^n * x^n / (1 + (n+2)*x)^n.

%I A229039 #24 Mar 12 2024 02:42:15
%S A229039 1,3,7,24,108,600,3960,30240,262080,2540160,27216000,319334400,
%T A229039 4071513600,56043187200,828193766400,13076743680000,219689293824000,
%U A229039 3912561709056000,73627297615872000,1459741204905984000,30411275102208000000,664182248232222720000
%N A229039 G.f.: Sum_{n>=0} (n+2)^n * x^n / (1 + (n+2)*x)^n.
%C A229039 More generally, we have the identity:
%C A229039 if Sum_{n>=0} a(n)*x^n = Sum_{n>=0} (b*n+c)^n * x^n / (1 + (b*n+c)*x)^n,
%C A229039 then Sum_{n>=0} a(n)*x^n/n! = (2 - 2*(b-c)*x + b*(b-2*c)*x^2)/(2*(1-b*x)^2)
%C A229039 so that a(n) = (b*n + (b+2*c)) * b^(n-1) * n!/2 for n>0 with a(0)=1.
%F A229039 a(n) = (n+5) * n!/2 for n>0 with a(0)=1.
%F A229039 E.g.f.: (2 + 2*x - 3*x^2)/(2*(1-x)^2).
%F A229039 From _Amiram Eldar_, Dec 11 2022: (Start)
%F A229039 Sum_{n>=0} 1/a(n) = 18*e - 237/5.
%F A229039 Sum_{n>=0} (-1)^n/a(n) = 243/5 - 130/e. (End)
%e A229039 O.g.f.: A(x) = 1 + 3*x + 7*x^2 + 24*x^3 + 108*x^4 + 600*x^5 + 3960*x^6 +...
%e A229039 where
%e A229039 A(x) = 1 + 3*x/(1+3*x) + 4^2*x^2/(1+4*x)^2 + 5^3*x^3/(1+5*x)^3 + 6^4*x^4/(1+6*x)^4 + 7^5*x^5/(1+7*x)^5 +...
%e A229039 E.g.f.: E(x) = 1 + 3*x + 7*x^2/2! + 24*x^3/3! + 108*x^4/4! + 600*x^5/5! +...
%e A229039 where
%e A229039 E(x) = 1 + 3*x + 7/2*x^2 + 4*x^3 + 9/2*x^4 + 5*x^5 + 11/2*x^6 + 6*x^7 +...
%e A229039 which is the expansion of: (2 + 2*x - 3*x^2) / (2 - 4*x + 2*x^2).
%t A229039 a[n_] := (n + 5)*n!/2; a[0] = 1; Array[a, 20, 0] (* _Amiram Eldar_, Dec 11 2022 *)
%o A229039 (PARI) {a(n)=polcoeff( sum(m=0, n, ((m+2)*x)^m / (1 + (m+2)*x +x*O(x^n))^m), n)}
%o A229039 for(n=0, 20, print1(a(n), ", "))
%o A229039 (PARI) {a(n)=if(n==0,1, (n+5) * n!/2 )}
%o A229039 for(n=0, 20, print1(a(n), ", "))
%Y A229039 Cf. A038720, A230056,  A187735, A187738, A187739, A229039, A221160, A221161, A187740.
%K A229039 nonn
%O A229039 0,2
%A A229039 _Paul D. Hanna_, Sep 11 2013