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%I #47 Jul 10 2024 09:37:17
%S 1,5,25,105,420,1596,5880,21120,74415,258115,883883,2994355,10051860,
%T 33479460,110750580,364177332,1191186855,3877914915,12571302975,
%U 40598200335,130657125984,419173385400,1340928798300,4278305877300,13617034683525,43243221276801,137040737988105
%N Expansion of 1/(1 - 2*x - 3*x^2)^(5/2).
%C From _Petros Hadjicostas_, Jun 03 2020: (Start)
%C For n >= 4, 2*a(n-4) counts 3-sets of leaves in "0,1,2" Motzkin rooted trees with n edges. "0,1,2" trees are rooted trees where each vertex has out-degree zero, one, or two. They are counted by the Motzkin numbers A001006.
%C For "0,1,2" trees, Salaam (2008) proved that the g.f. of the number of r-sets of leaves is A000108(r-1) * z^(2*r-2) * T(z)^(2*r-1), where T(z) = 1/sqrt(1 - 2*z - 3*z^2) is the g.f. of the central trinomial numbers A002426.
%C For r = 2, we get a shifted version of A102839. For r = 3, we get twice of a shifted version of the current sequence. (End)
%H Vincenzo Librandi, <a href="/A245551/b245551.txt">Table of n, a(n) for n = 0..1000</a>
%H Robert Coquereaux and Jean-Bernard Zuber, <a href="https://arxiv.org/abs/2305.01100">Counting partitions by genus. II. A compendium of results</a>, arXiv:2305.01100 [math.CO], 2023. See p. 6.
%H Lifoma Salaam, <a href="https://search.proquest.com/docview/193997569">Combinatorial statistics on phylogenetic trees</a>, Ph.D. Dissertation, Howard University, Washington D.C., 2008; see Theorem 39 (p. 25).
%H J. Y. X. Yang, M. X. X. Zhong, and R. D. P. Zhou, <a href="http://arxiv.org/abs/1406.2583">On the Enumeration of (s, s+ 1, s+2)-Core Partitions</a>, arXiv preprint arXiv:1406.2583 [math.CO], 2014. See Theorem 4.2.
%F a(n) ~ 3^(n+3/2) * n^(3/2) / (8*sqrt(Pi)). - _Vaclav Kotesovec_, Jul 31 2014
%F a(n) = (2+3/n)*a(n-1) + (3+9/n)*a(n-2) for n >= 2. - _Robert Israel_, Aug 01 2014
%F a(n) = (binomial(n+4,2)/6) * Sum_{k=0..floor(n/2)} binomial(n+2,n-2*k) * binomial(2*k+2,k). - _Seiichi Manyama_, Jul 10 2024
%e From _Petros Hadjicostas_, Jun 03 2020: (Start)
%e Out of the A001006(4) = 9 Motzkin trees with n = 4 edges, only the following 2*a(4-4) = 2 have 3-sets of leaves:
%e A A
%e / \ / \
%e / \ / \
%e B C B C
%e / \ / \
%e / \ / \
%e D E D E
%e {C, D, E} {B, D, E}
%e (End)
%p A[0]:= 1: A[1]:= 5:
%p for n from 2 to 100 do
%p A[n]:= (2+3/n)*A[n-1] + (3+9/n)*A[n-2]
%p od:
%p seq(A[n],n=0..100); # _Robert Israel_, Aug 01 2014
%t CoefficientList[Series[1/(1 - 2 x - 3 x^2)^(5/2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 01 2014 *)
%o (PARI) x='x+O('x^50); Vec(1/(1-2*x-3*x^2)^(5/2)) \\ _G. C. Greubel_, Apr 06 2017
%Y Cf. A000108, A001006, A002426, A102839.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Jul 30 2014
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