A207833 E.g.f.: T(T(x)), where T(x) is the e.g.f. for labeled rooted trees, A000169.
1, 4, 30, 332, 4880, 89742, 1986124, 51471800, 1530489744, 51395228090, 1924687118684, 79553145323940, 3598161485778808, 176797212122233094, 9378715234039802340, 534259395682874552048, 32528761111972930621472, 2108146039402630977388530, 144899759883703796130871468, 10528261771566724089621962780
Offset: 1
Keywords
Examples
E.g.f.: A(x) = x + 4*x^2/2! + 30*x^3/3! + 332*x^4/4! + 4880*x^5/5! +... Euler's tree function T(x) satisfies: T(x/exp(x)) = x, and begins: T(x) = x + 2*x^2/2! + 3^2*x^3/3! + 4^3*x^4/4! + 5^4*x^5/5! +...+ A000169(n)*x^n/n! +... where e.g.f. A(x) = T(T(x)).
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..200
- Yoshida Tomoyuki, Categorical aspects of generating functions. I. Exponential formulas and Krull-Schmidt categories, J. Algebra 240 (2001), no. 1, 40-82. MR1830543 (2002e:18008). See Sect. 6.8.
Programs
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Mathematica
nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[ 0,nn]!CoefficientList[ ComposeSeries[ Series[t,{x,0,nn}],Series[t,{x,0,nn}]],x] (* Geoffrey Critzer, Sep 16 2012 *) Rest[CoefficientList[Series[-LambertW[LambertW[-x]], {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Feb 24 2014 *) -
PARI
{a(n)=if(n==0||n==1, 1, n^(n-1)-sum(k=1, n-1, (-1)^(n-k)*binomial(n, k)*k^(n-k)*a(k)))} \\ Paul D. Hanna, Nov 21 2012
Formula
a(n) = 1/n * Sum_{k=1..n} C(n,k)*k^k*n^(n-k). [Vladimir Kruchinin, Sep 24 2012]
a(n) = n^(n-1) - Sum_{k=1..n-1} (-1)^(n-k) * C(n, k) * k^(n-k) * a(k) for n>1 with a(1)=1. - Paul D. Hanna, Nov 21 2012
E.g.f. A(x) satisfies: A(x) = Sum_{n>=1} n^(n-1)*T(x)^n/n!, by definition.
E.g.f. A(x) satisfies: A(x/exp(x)) = T(x) = Sum_{n>=1} n^(n-1)*x^n/n!. - Paul D. Hanna, Jul 04 2013
a(n) ~ n^(n-1) * exp(n*exp(-1)) / sqrt(1-exp(-1)). - Vaclav Kotesovec, Feb 24 2014
Comments