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

1,2

Comments

Exponential series reversal gives A185298 with alternating signs: 1, -4, 18, -92, 520, ... . - Vladimir Reshetnikov, Aug 04 2019

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, A185298, 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