A050353 Number of 5-level labeled linear rooted trees with n leaves.

1, 1, 9, 121, 2169, 48601, 1306809, 40994521, 1469709369, 59277466201, 2656472295609, 130952452264921, 7042235448544569, 410269802967187801, 25740278881968596409, 1730295054262416751321, 124066865052334175027769

Offset

0,3

Links

Michael De Vlieger, Table of n, a(n) for n = 0..360

Norihiro Nakashima, Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Index entries for sequences related to rooted trees

Formula

E.g.f.: (4 - 3*exp(x))/(5 - 4*exp(x)).

a(n) is asymptotic to (1/20)*n!/log(5/4)^(n+1). More generally if m>1, the number of m-level labeled linear rooted trees with n leaves is asymptotic to n!/log(m/(m-1))^(n+1)/(m^2-m). - Benoit Cloitre, Jan 30 2003

For m-level trees (m>1), e.g.f. is (m-1-(m-2)*e^x)/(m-(m-1)*e^x) and number of trees is 1/(m*(m-1))*sum(k>=0, (1-1/m)^k*k^n). Here m=5, so a(n)=(1/20)*sum(k>=0, (4/5)^k*k^n) (for n>0). - Benoit Cloitre, Jan 30 2003

Let f(x) = (1+x)*(1+2*x). Let D be the operator g(x) -> d/dx(f(x)*g(x)). Then for n>=1, a(n) = D^(n-1)(1) evaluated at x = 3/2. Compare with the result A000670(n) = D^(n-1)(1) at x = 0. See also A194649. - Peter Bala, Sep 05 2011

E.g.f.: 1 + x/(G(0)-5*x) where G(k)= x + k + 1 - x*(k+1)/G(k+1); (continued fraction Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Jul 11 2012

a(n) = (1/20) * Sum_{k>=1} k^n * (4/5)^k for n>0. - Paul D. Hanna, Nov 28 2014

a(n) = Sum_{k=1..n} Stirling2(n, k) * k! * 4^(k-1). - Paul D. Hanna, Nov 28 2014, after Vladeta Jovovic in A050351

a(n) = 1 + 4 * Sum_{k=1..n-1} binomial(n,k) * a(k). - Ilya Gutkovskiy, Jun 08 2020

Maple

seq(coeff(series( (4-3*exp(x))/(5-4*exp(x)), x, n+1)*n!, x, n), n = 0..20); # G. C. Greubel, Jun 08 2020

Mathematica

max = 16; f[x_] := (4-3*E^x) / (5-4*E^x); CoefficientList[ Series[ f[x], {x, 0, max}], x]*Range[0, max]! (* Jean-François Alcover, Nov 14 2011, after g.f. *)

Prog

(PARI) a(n)=n!*if(n<0, 0, polcoeff((4-3*exp(x))/(5-4*exp(x))+O(x^(n+1)), n))
(PARI) {a(n)=if(n==0, 1, (1/20)*round(suminf(k=1, k^n * (4/5)^k *1.)))} \\ Paul D. Hanna, Nov 28 2014
(Magma) R<x>:=PowerSeriesRing(Rationals(), 20); Coefficients(R!(Laplace( (4-3*Exp(x))/(5-4*Exp(x)) ))); // G. C. Greubel, Jun 08 2020
(Sage) [1]+[sum( 4^(j-1)*factorial(j)*stirling_number2(n, j) for j in (1..n)) for n in (1..20)] # G. C. Greubel, Jun 08 2020

Crossrefs

Cf. A000670, A050351 - A050359.

Equals 1/4 * A094417(n) for n>0.

Sequence in context: A246467 A202835 A321847 * A112941 A352119 A258380

Adjacent sequences: A050350 A050351 A050352 * A050354 A050355 A050356

Keyword

nonn

Author

Christian G. Bower, Oct 15 1999

Status

approved