A161635 E.g.f. satisfies A(x) = exp( x/(1 - x*A(x))^2 ).

1, 1, 5, 43, 553, 9501, 204961, 5330599, 162432593, 5677941817, 224018814241, 9848702243931, 477481361216377, 25309471236379669, 1456206709854725921, 90387017392004356591, 6020486941130334199201, 428348710658269120403313

Offset: 0

Author

Paul D. Hanna, Jun 19 2009

Keywords

• nonn

Examples

			E.g.f: A(x) = 1 + x + 5*x^2/2! + 43*x^3/3! + 553*x^4/4! + 9501*x^5/5! +...
log(A(x))/x = 1 + 2*x*A(x) + 3*x^2*A(x)^2 + 4*x^3*A(x)^3 + 5*x^4*A(x)^4 +...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..360

Crossrefs

Cf. A161630, A364980.

Programs

Mathematica

Flatten[{1,Table[n!*Sum[(n-k+1)^(k-1)/k! * Binomial[n+k-1,n-k],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jan 10 2014 *)
FindRoot[{2*s*r^2 == (1-r*s)^3, r == Log[s]*(1-r*s)^2},{r,1/2},{s,1}, WorkingPrecision->50] (* program for numerical values of constants r and s, Vaclav Kotesovec, Jan 10 2014 *)

PARI

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

PARI

{a(n,m=1)=my(A=1+x+x*O(x^n));for(i=1,n,A=exp(x/(1-x*A)^2));n!*polcoeff(A^m,n)}

Formula

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

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

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

...

E.g.f.: A(x) = (1/x)*Series_Reversion[ (1-x)^2*LambertW(x/(1-x)^2) ].

a(n) ~ sqrt(s*(1+r*s)/(3+2*r-6*r*s+3*r^2*s^2)) * n^(n-1) / (exp(n) * r^n), where r = 0.219092370374602436272454670371..., s = 1.952248277910295452167538973654... are the roots of the equations 2*s*r^2 = (1-r*s)^3 and r = log(s) * (1-r*s)^2. - Vaclav Kotesovec, Jan 10 2014