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A023426 a(n) = a(n-1) + Sum_{k=0..n-4} a(k)*a(n-4-k), a(0) = 1. Generalized Catalan Numbers.

%I A023426 #50 Jul 13 2024 04:45:53
%S A023426 1,1,1,1,2,4,7,11,18,32,59,107,191,343,627,1159,2146,3972,7373,13757,
%T A023426 25781,48437,91165,171945,325096,616066,1169667,2224355,4236728,
%U A023426 8082374,15441719,29542411,56590472,108532322,208387711,400551615,770710831,1484383399
%N A023426 a(n) = a(n-1) + Sum_{k=0..n-4} a(k)*a(n-4-k), a(0) = 1. Generalized Catalan Numbers.
%C A023426 Number of lattice paths from (0,0) to (n,0) that stay weakly in the first quadrant and such that each step is either U=(2,1),D=(2,-1), or H=(1,0). E.g. a(5)=4 because we have HHHHH, HUD, UDH and UHD. - _Emeric Deutsch_, Dec 23 2003
%C A023426 Hankel transform is A132380(n+3). - _Paul Barry_, May 22 2009
%H A023426 Vincenzo Librandi, <a href="/A023426/b023426.txt">Table of n, a(n) for n = 0..1000</a>
%H A023426 Andrei Asinowski, Cyril Banderier and Valerie Roitner, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/several_patterns.pdf">Generating functions for lattice paths with several forbidden patterns</a>, (2019).
%H A023426 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. 10, 19-21.
%H A023426 K. Park and G.S. Cheon, <a href="http://www.kms.or.kr/conference/abstract/search_view.html?num=5098&amp;uid=32">Lattice path counting with a bounded strip restriction</a>
%F A023426 G.f.: [1-z-sqrt((1-z)^2-4z^4)]/[2z^4]. - _Emeric Deutsch_, Dec 23 2003
%F A023426 From _Paul Barry_, May 22 2009: (Start)
%F A023426 G.f.: 1/(1-x-x^4/(1-x-x^4/(1-x-x^4/(1-x-x^4/(1-... (continued fraction).
%F A023426 G.f.: (1/(1-x))c(x^4/(1-x)^2), c(x) the g.f. of A000108.
%F A023426 a(n) = Sum_{k=0..floor(n/4)} C(n-2k,2k)*A000108(k). (End)
%F A023426 D-finite with recurrence (n+4)*a(n) +(n+4)*a(n-1) -(5*n+8)*a(n-2) +3*n*a(n-3) +4*(2-n)*a(n-4) +12*(3-n)*a(n-5)=0. - _R. J. Mathar_, Sep 29 2012
%F A023426 a(n) ~ sqrt(3) * 2^(n+3/2) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Feb 01 2014
%F A023426 G.f. A(x) satisfies: A(x) = (1 + x^4 * A(x)^2) / (1 - x). - _Ilya Gutkovskiy_, Jul 20 2021
%F A023426 a(n) = hypergeom([(1 - n)/4, (2 - n)/4, (3 - n)/4, -n/4], [2, (1 - n)/2, -n/2], 64). - _Peter Luschny_, Jul 12 2024
%p A023426 a := n -> hypergeom([(1 - n)/4, (2 - n)/4, (3 - n)/4, -n/4], [2, (1 - n)/2, -n/2], 2^6): seq(simplify(a(n)), n = 0..35); # _Peter Luschny_, Jul 12 2024
%t A023426 Clear[ a ]; a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-4-k ], {k, 0, n-4} ];
%t A023426 CoefficientList[Series[(1-x-Sqrt[(1-x)^2-4*x^4])/(2*x^4), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 01 2014 *)
%Y A023426 Cf. A000108, A001006, A004148, A006318.
%K A023426 nonn,easy
%O A023426 0,5
%A A023426 _Olivier Gérard_
%E A023426 Name extended by a formula from the author in Mathematica by _Peter Luschny_, Jul 13 2024