A050352 Number of 4-level labeled linear rooted trees with n leaves.

1, 1, 7, 73, 1015, 17641, 367927, 8952553, 248956855, 7788499561, 270732878647, 10351919533033, 431806658432695, 19512813265643881, 949587798053709367, 49512355251796513513, 2753726282896986372535, 162725978752448205162601

Offset: 0

Author

Christian G. Bower, Oct 15 1999

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..370
• Marian Muresan, A concrete approach to classical analysis, CMS Books in Mathematics (2009) Table 10.2.
• 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

Crossrefs

Cf. A000670, A050351 - A050359.

Equals 1/3 * A032033(n) for n>0.

Programs

Magma

[1] cat [(&+[3^(j-1)*Factorial(j)*StirlingSecond(n,j): j in [1..n]]) : n in [0..20]]; // G. C. Greubel, Jun 08 2020

Maple

seq(coeff(series( (3-2*exp(x))/(4-3*exp(x)), x, n+1)*n!, x, n), n = 0..20); # G. C. Greubel, Jun 08 2020

Mathematica

With[{nn=20}, CoefficientList[Series[(3-2Exp[x])/(4-3Exp[x]),{x,0,nn}], x]*Range[0,nn]!] (* Harvey P. Dale, Aug 16 2012 *)

PARI

a(n)=n!*if(n<0,0,polcoeff((3-2*exp(x))/(4-3*exp(x))+O(x^(n+1)),n))

PARI

{a(n)=if(n==0, 1, (1/12)*round(suminf(k=1, k^n * (3/4)^k *1.)))} \\ Paul D. Hanna, Nov 28 2014

Sage

[1]+[sum( 3^(j-1)*factorial(j)*stirling_number2(n,j) for j in (1..n)) for n in (1..20)] # G. C. Greubel, Jun 08 2020

Formula

E.g.f.: (3 - 2*exp(x))/(4 - 3*exp(x)).

a(n) is asymptotic to (1/12)*n!/log(4/3)^(n+1). - 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=4, so a(n)=(1/12)*sum(k>=0, (3/4)^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 = 1. 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)-4*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) = (1/12) * Sum_{k>=1} k^n * (3/4)^k for n>0. - Paul D. Hanna, Nov 28 2014

a(n) = Sum_{k=1..n} Stirling2(n, k) * k! * 3^(k-1). - Paul D. Hanna, Nov 28 2014, after Vladeta Jovovic in A050351

a(n) = 1 + 3 * Sum_{k=1..n-1} binomial(n,k) * a(k). - Ilya Gutkovskiy, Jun 08 2020