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%I #43 Sep 08 2022 08:44:58
%S 1,1,7,73,1015,17641,367927,8952553,248956855,7788499561,270732878647,
%T 10351919533033,431806658432695,19512813265643881,949587798053709367,
%U 49512355251796513513,2753726282896986372535,162725978752448205162601
%N Number of 4-level labeled linear rooted trees with n leaves.
%H G. C. Greubel, <a href="/A050352/b050352.txt">Table of n, a(n) for n = 0..370</a>
%H Marian Muresan, <a href="https://doi.org/10.1007/978-0-387-78933-0">A concrete approach to classical analysis</a>, CMS Books in Mathematics (2009) Table 10.2.
%H Norihiro Nakashima, Shuhei Tsujie, <a href="https://arxiv.org/abs/1904.09748">Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species</a>, arXiv:1904.09748 [math.CO], 2019.
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F E.g.f.: (3 - 2*exp(x))/(4 - 3*exp(x)).
%F a(n) is asymptotic to (1/12)*n!/log(4/3)^(n+1). - _Benoit Cloitre_, Jan 30 2003
%F 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
%F 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
%F 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
%F a(n) = (1/12) * Sum_{k>=1} k^n * (3/4)^k for n>0. - _Paul D. Hanna_, Nov 28 2014
%F a(n) = Sum_{k=1..n} Stirling2(n, k) * k! * 3^(k-1). - _Paul D. Hanna_, Nov 28 2014, after _Vladeta Jovovic_ in A050351
%F a(n) = 1 + 3 * Sum_{k=1..n-1} binomial(n,k) * a(k). - _Ilya Gutkovskiy_, Jun 08 2020
%p 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
%t With[{nn=20}, CoefficientList[Series[(3-2Exp[x])/(4-3Exp[x]),{x,0,nn}], x]*Range[0,nn]!] (* _Harvey P. Dale_, Aug 16 2012 *)
%o (PARI) a(n)=n!*if(n<0,0,polcoeff((3-2*exp(x))/(4-3*exp(x))+O(x^(n+1)),n))
%o (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
%o (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
%o (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
%Y Cf. A000670, A050351 - A050359.
%Y Equals 1/3 * A032033(n) for n>0.
%K nonn
%O 0,3
%A _Christian G. Bower_, Oct 15 1999
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