A161631 - OEIS

%I #15 Dec 02 2023 13:30:49

%S 1,1,2,9,52,425,4206,50827,713000,11500785,208833850,4226139731,

%T 94226705772,2296472176297,60727113115046,1732020500240955,

%U 52998549321251536,1731977581804704737,60205422811336194546

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

%H Seiichi Manyama, <a href="/A161631/b161631.txt">Table of n, a(n) for n = 0..402</a>

%F E.g.f.: A(x) = 1 - LambertW(-x^2*exp(x))/x.

%F E.g.f.: A(x) = 1 + Sum_{n>=1} x^(2*n-1) * n^(n-1) * exp(n*x) / n!.

%F E.g.f.: A(x) = (1/x)*Series_Reversion(x/B(x)) where B(x) = 1 + x*exp(x)/B(x) = (1+sqrt(1+4*x*exp(x)))/2.

%F a(n) = n*A125500(n-1) for n>0, where exp(x*A(x)) = e.g.f. of A125500.

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

%F a(n) ~ sqrt(1+LambertW(1/(2*exp(1/2)))) * n^(n-1) / (exp(n) * 2^(n+1/2) * (LambertW(1/(2*exp(1/2))))^(n+1)). - _Vaclav Kotesovec_, Jul 09 2013

%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 425*x^5/5! +...

%e exp(x*A(x)) = 1 + x + 3*x^2/2! + 13*x^3/3! + 85*x^4/4! + 701*x^5/5! +...

%t CoefficientList[Series[1-LambertW[-x^2*E^x]/x, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jul 09 2013 *)

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

%Y Cf. A125500, A364978, A364979.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jun 18 2009