OFFSET
0,5
COMMENTS
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
Hankel transform is A132380(n+3). - Paul Barry, May 22 2009
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
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
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
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Name extended by a formula from the author in Mathematica by Peter Luschny, Jul 13 2024
STATUS
approved