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A207833 E.g.f.: T(T(x)), where T(x) is the e.g.f. for labeled rooted trees, A000169. 5
1, 4, 30, 332, 4880, 89742, 1986124, 51471800, 1530489744, 51395228090, 1924687118684, 79553145323940, 3598161485778808, 176797212122233094, 9378715234039802340, 534259395682874552048, 32528761111972930621472, 2108146039402630977388530, 144899759883703796130871468, 10528261771566724089621962780 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

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

EXAMPLE

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)).

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 *)

PROG

(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

CROSSREFS

Cf. A000169, A227278, A227176.

Sequence in context: A211828 A277759 A006149 * A121413 A001761 A292220

Adjacent sequences:  A207830 A207831 A207832 * A207834 A207835 A207836

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Feb 20 2012

STATUS

approved

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Last modified February 18 05:48 EST 2018. Contains 299298 sequences. (Running on oeis4.)