This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A245551 #52 Aug 23 2025 09:50:32
%S A245551 1,5,25,105,420,1596,5880,21120,74415,258115,883883,2994355,10051860,
%T A245551 33479460,110750580,364177332,1191186855,3877914915,12571302975,
%U A245551 40598200335,130657125984,419173385400,1340928798300,4278305877300,13617034683525,43243221276801,137040737988105
%N A245551 Expansion of 1/(1 - 2*x - 3*x^2)^(5/2).
%C A245551 From _Petros Hadjicostas_, Jun 03 2020: (Start)
%C A245551 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 A245551 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 A245551 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 A245551 Vincenzo Librandi, <a href="/A245551/b245551.txt">Table of n, a(n) for n = 0..1000</a>
%H A245551 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 A245551 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 A245551 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 A245551 a(n) ~ 3^(n+3/2) * n^(3/2) / (8*sqrt(Pi)). - _Vaclav Kotesovec_, Jul 31 2014
%F A245551 a(n) = (2+3/n)*a(n-1) + (3+9/n)*a(n-2) for n >= 2. - _Robert Israel_, Aug 01 2014
%F A245551 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
%F A245551 a(n) = Sum_{k=0..n} (-2)^k * (3/2)^(n-k) * binomial(-5/2,k) * binomial(k,n-k). - _Seiichi Manyama_, Aug 23 2025
%e A245551 From _Petros Hadjicostas_, Jun 03 2020: (Start)
%e A245551 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 A245551 A A
%e A245551 / \ / \
%e A245551 / \ / \
%e A245551 B C B C
%e A245551 / \ / \
%e A245551 / \ / \
%e A245551 D E D E
%e A245551 {C, D, E} {B, D, E}
%e A245551 (End)
%p A245551 A[0]:= 1: A[1]:= 5:
%p A245551 for n from 2 to 100 do
%p A245551 A[n]:= (2+3/n)*A[n-1] + (3+9/n)*A[n-2]
%p A245551 od:
%p A245551 seq(A[n],n=0..100); # _Robert Israel_, Aug 01 2014
%t A245551 CoefficientList[Series[1/(1 - 2 x - 3 x^2)^(5/2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 01 2014 *)
%o A245551 (PARI) x='x+O('x^50); Vec(1/(1-2*x-3*x^2)^(5/2)) \\ _G. C. Greubel_, Apr 06 2017
%Y A245551 Cf. A000108, A001006, A002426, A102839, A374506.
%K A245551 nonn,changed
%O A245551 0,2
%A A245551 _N. J. A. Sloane_, Jul 30 2014