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%I #57 Apr 20 2024 09:53:57
%S 1,0,0,1,2,4,8,17,38,88,208,498,1204,2936,7216,17861,44486,111408,
%T 280352,708526,1797564,4576472,11688496,29939786,76894684,197974480,
%U 510864480,1321031716,3422685992,8884010928,23098674400,60152509613,156879556678
%N a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-1)*a(1) for n >= 5, with a(1) = 1, a(2) = a(3) = 0, a(4) = 1.
%C Number of lattice paths from (0,0) to (n-4,0) that stay weakly in the first quadrant and such that each step is either U=(2,1), D=(2,-1), blue H=(1,0), or red h=(1,0) (n>=4). E.g., a(8)=17 because we have 16 horizontal paths of length 4 with all combinations of blue and red (1,0) steps and, in addition, UD. - _Emeric Deutsch_, Dec 23 2003
%C From _Ricardo Gómez Aíza_, Mar 20 2024: (Start)
%C a(n+3) is the number of rooted plane 2-trees with nonempty integer compositions labeling all the nodes, including the root, with total size n >= 0. The total size is the number of edges in the tree plus the sum of the sizes of the integer compositions labeling all the nodes.
%C Examples: a(3)=0 because there are no elements of size zero; a(4)=1, a(5)=2, a(6)=4 and a(7)=8 because in each case, the elements are trees that consist of the root alone labeled with the compositions of 1, 2, 3 and 4, respectively; a(8)=17 because now we have 17 elements of size 5, the first 16 coming from the root alone labeled with the compositions of 5, plus the 2-tree that consists of the root with two descendants, with each of the three nodes labeled with the composition 1=1. (End)
%H Reinhard Zumkeller, <a href="/A025276/b025276.txt">Table of n, a(n) for n = 1..1000</a>
%H Ricardo Gómez Aíza, <a href="https://arxiv.org/abs/2402.16111">Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis</a>, arXiv:2402.16111 [math.CO], 2024. See pp. 20.
%H Yan Zhuang, <a href="https://arxiv.org/abs/1508.02793">A generalized Goulden-Jackson cluster method and lattice path enumeration</a>, Discrete Mathematics 341.2 (2018): 358-379; arXiv:1508.02793 [math.CO], 2015-2018.
%F G.f.: (1 - sqrt((1-2*z)^2 - 4*z^4))/2. - _Emeric Deutsch_, Dec 23 2003
%F Recurrence: n*a(n) = 2*(2*n-3)*a(n-1) - 4*(n-3)*a(n-2) + 4*(n-6)*a(n-4). - _Vaclav Kotesovec_, Jan 25 2015
%F a(n) ~ sqrt((9-5*sqrt(3))/(8*Pi*n^3))*(2/(sqrt(3)-1))^n. - _Ricardo Gómez Aíza_, Mar 01 2024
%t nmax = 30; aa = ConstantArray[0,nmax]; aa[[1]] = 1; aa[[2]] = 0; aa[[3]] = 0; aa[[4]] = 1; Do[aa[[n]] = Sum[aa[[k]]*aa[[n-k]],{k,1,n-1}],{n,5,nmax}]; aa (* _Vaclav Kotesovec_, Jan 25 2015 *)
%o (Haskell)
%o a025276 n = a025276_list !! (n-1)
%o a025276_list = 1 : 0 : 0 : 1 : f [1,0,0,1] where
%o f xs = x' : f (x':xs) where
%o x' = sum $ zipWith (*) xs a025276_list
%o -- _Reinhard Zumkeller_, Nov 03 2011
%K nonn,changed
%O 1,5
%A _Clark Kimberling_
%E Definition improved by _Bernard Schott_, Jun 27 2022
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