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%I #26 Dec 23 2022 07:50:26
%S 1,3,16,120,1152,13440,184320,2903040,51609600,1021870080,22295347200,
%T 531372441600,13733933875200,382588157952000,11426632984166400,
%U 364223926370304000,12340763622899712000,442896294466289664000,16783438527143608320000
%N G.f.: Sum_{n>=0} (2*n+1)^n * x^n / (1 + (2*n+1)*x)^n.
%C Compare g.f. to the identity (cf. A001710):
%C Sum_{n>=0} n^n * x^n / (1 + n*x)^n = 1 + Sum_{n>=1} (n+1)!/2 * x^n.
%C More generally,
%C if Sum_{n>=0} a(n)*x^n = Sum_{n>=0} (b*n+c)^n * x^n / (1 + (b*n+c)*x)^n,
%C 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 so that a(n) = (b*n + (b+2*c)) * b^(n-1) * n!/2 for n>0 with a(0)=1.
%F a(n) = (n+2) * 2^(n-1) * n!.
%F E.g.f.: (1-x)/(1-2*x)^2.
%F From _Amiram Eldar_, Dec 23 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 8 - 4*sqrt(e).
%F Sum_{n>=0} (-1)^n/a(n) = 8 - 12/sqrt(e). (End)
%e G.f.: A(x) = 1 + 3*x + 16*x^2 + 120*x^3 + 1152*x^4 + 13440*x^5 +...
%e where
%e A(x) = 1 + 3*x/(1+3*x) + 5^2*x^2/(1+5*x)^2 + 7^3*x^3/(1+7*x)^3 + 9^4*x^4/(1+9*x)^4 + 11^5*x^5/(1+11*x)^5 +...
%t a[n_] := (n + 2)*2^(n - 1)*n!; Array[a, 20, 0] (* _Amiram Eldar_, Dec 23 2022 *)
%o (PARI) {a(n)=polcoeff( sum(m=0,n,((2*m+1)*x)^m / (1 + (2*m+1)*x +x*O(x^n))^m),n)}
%o for(n=0,20,print1(a(n),", "))
%o (PARI) {a(n) = (n+2)*2^(n-1)*n!}
%Y Cf. A001710, A014479, A187738, A187739, A221160, A221161, A187740.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 02 2013
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