A023426 a(n) = a(n-1) + Sum_{k=0..n-4} a(k)*a(n-4-k), a(0) = 1. Generalized Catalan Numbers.
1, 1, 1, 1, 2, 4, 7, 11, 18, 32, 59, 107, 191, 343, 627, 1159, 2146, 3972, 7373, 13757, 25781, 48437, 91165, 171945, 325096, 616066, 1169667, 2224355, 4236728, 8082374, 15441719, 29542411, 56590472, 108532322, 208387711, 400551615, 770710831, 1484383399
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Andrei Asinowski, Cyril Banderier and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
- Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See pp. 10, 19-21.
- K. Park and G.S. Cheon, Lattice path counting with a bounded strip restriction
Programs
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Maple
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
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Mathematica
Clear[ a ]; a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-4-k ], {k, 0, n-4} ]; CoefficientList[Series[(1-x-Sqrt[(1-x)^2-4*x^4])/(2*x^4), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2014 *)
Formula
G.f.: [1-z-sqrt((1-z)^2-4z^4)]/[2z^4]. - Emeric Deutsch, Dec 23 2003
From Paul Barry, May 22 2009: (Start)
G.f.: 1/(1-x-x^4/(1-x-x^4/(1-x-x^4/(1-x-x^4/(1-... (continued fraction).
G.f.: (1/(1-x))c(x^4/(1-x)^2), c(x) the g.f. of A000108.
a(n) = Sum_{k=0..floor(n/4)} C(n-2k,2k)*A000108(k). (End)
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
a(n) ~ sqrt(3) * 2^(n+3/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 01 2014
G.f. A(x) satisfies: A(x) = (1 + x^4 * A(x)^2) / (1 - x). - Ilya Gutkovskiy, Jul 20 2021
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
Extensions
Name extended by a formula from the author in Mathematica by Peter Luschny, Jul 13 2024
Comments