A161632 - OEIS

%I #11 Nov 02 2024 09:13:10

%S 1,2,6,42,392,4970,78492,1489838,33105648,842437170,24181696820,

%T 772887702422,27228973364232,1048392980781770,43802436902618604,

%U 1973819502540516990,95426799849067842272,4927195390491532227170

%N E.g.f. satisfies A(x) = (1 + x*exp(x*A(x)))^2.

%F a(n) = n!*Sum_{k=0..n} C(2*(n-k+1),k)/(n-k+1) * k^(n-k)/(n-k)!.

%F If A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! then

%F a(n,m) = n!*Sum_{k=0..n} m*C(2*(n-k+m),k)/(n-k+m) * k^(n-k)/(n-k)!.

%F E.g.f.: A(x) = (1/x)*Series_Reversion(x/B(x)) where B(x) = (1 + x*exp(x)/B(x))^2.

%F a(n) ~ sqrt(2*s^(3/2)*(2-5*sqrt(s)+3*s)/(2*sqrt(s)-1)) * (2*s-2*sqrt(s))^n * n^(n-1) / exp(n), where s = 3.533778497303240223520495... is the root of the equation (2-2/sqrt(s)) * log(2*(sqrt(s)-2*s+s^(3/2))) = 1. - _Vaclav Kotesovec_, Jan 10 2014

%F E.g.f.: B(x)^2, where B(x) is the e.g.f. of A364978. - _Seiichi Manyama_, Nov 02 2024

%e E.g.f.: A(x) = 1 + 2*x + 6*x^2/2! + 42*x^3/3! + 392*x^4/4! + 4970*x^5/5! +...

%e A(x)^(1/2) = 1 + x + 2*x^2/2! + 15*x^3/3! + 124*x^4/4! + 1565*x^5/5! +...

%t Flatten[{1,Table[n!*Sum[Binomial[2*(n-k+1),k]/(n-k+1) * k^(n-k)/(n-k)!,{k,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, Jan 10 2014 *)

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

%Y Cf. A161631, A364978.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 18 2009