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A050352 Number of 4-level labeled linear rooted trees with n leaves. 11
1, 1, 7, 73, 1015, 17641, 367927, 8952553, 248956855, 7788499561, 270732878647, 10351919533033, 431806658432695, 19512813265643881, 949587798053709367, 49512355251796513513, 2753726282896986372535, 162725978752448205162601 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..17.

Index entries for sequences related to rooted trees

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

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 *)

PROG

(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

CROSSREFS

Cf. A000670, A050351-A050359.

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

Sequence in context: A114429 A124547 A084363 * A261783 A250917 A112939

Adjacent sequences:  A050349 A050350 A050351 * A050353 A050354 A050355

KEYWORD

nonn

AUTHOR

Christian G. Bower, Oct 15 1999

STATUS

approved

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Last modified November 14 19:12 EST 2018. Contains 317214 sequences. (Running on oeis4.)