A161632 E.g.f. satisfies A(x) = (1 + x*exp(x*A(x)))^2.
1, 2, 6, 42, 392, 4970, 78492, 1489838, 33105648, 842437170, 24181696820, 772887702422, 27228973364232, 1048392980781770, 43802436902618604, 1973819502540516990, 95426799849067842272, 4927195390491532227170
Offset: 0
Keywords
Examples
E.g.f.: A(x) = 1 + 2*x + 6*x^2/2! + 42*x^3/3! + 392*x^4/4! + 4970*x^5/5! +... A(x)^(1/2) = 1 + x + 2*x^2/2! + 15*x^3/3! + 124*x^4/4! + 1565*x^5/5! +...
Programs
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Mathematica
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 *) -
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)!)}
Formula
a(n) = n!*Sum_{k=0..n} C(2*(n-k+1),k)/(n-k+1) * k^(n-k)/(n-k)!.
If A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! then
a(n,m) = n!*Sum_{k=0..n} m*C(2*(n-k+m),k)/(n-k+m) * k^(n-k)/(n-k)!.
E.g.f.: A(x) = (1/x)*Series_Reversion(x/B(x)) where B(x) = (1 + x*exp(x)/B(x))^2.
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
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A364978. - Seiichi Manyama, Nov 02 2024