A161633 E.g.f. satisfies A(x) = 1/(1 - x*exp(x*A(x))).
1, 1, 4, 27, 268, 3525, 57966, 1146061, 26500552, 702069129, 20974309210, 697754762001, 25584428686620, 1025230366195789, 44579963354153878, 2090676600895922565, 105191995364927688976, 5652501986238910061073, 323083811850594613809714, 19573120681427758058921881
Offset: 0
Keywords
Examples
E.g.f.: A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 268*x^4/4! + 3525*x^5/5! +... exp(x*A(x)) = 1 + x + 3*x^2/2! + 19*x^3/3! + 181*x^4/4! + 2321*x^5/5! +...
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..374
Crossrefs
Programs
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Mathematica
Flatten[{1,Table[n!*Sum[Binomial[n+1,k]/(n+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,binomial(n+m,k)*m/(n+m)*k^(n-k)/(n-k)!)
Formula
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = 1 + x*A(x)*exp(x*A(x)).
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(x)) ).
(3) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-1)*x*A(x)) for all fixed nonnegative m.
a(n) = n! * Sum_{k=0..n} binomial(n+1,k)/(n+1) * k^(n-k)/(n-k)!.
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! then a(n,m) = n! * Sum_{k=0..n} binomial(n+m,k)*m/(n+m) * k^(n-k)/(n-k)!.
a(n) ~ n^(n-1) * c * ((c-1)*c)^(n+1/2) / (sqrt(2*c-1) * exp(n)), where c = 1 + 1/(2*LambertW(1/2)) = 2.4215299358831166... - Vaclav Kotesovec, Jan 10 2014