A260772 Certain directed lattice paths.
1, 3, 10, 41, 190, 946, 4940, 26693, 147990, 837102, 4811860, 28027210, 165057100, 981177060, 5879570200, 35478788269, 215398416870, 1314794380374, 8064119033220, 49673222082782, 307163049317540, 1906066361809148, 11865666767361960, 74081851132379426
Offset: 0
Keywords
Links
- Lars Blomberg, Table of n, a(n) for n = 0..100
- M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
- M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, Discrete Mathematics, Volume 339, Issue 3, 6 March 2016, Pages 1116-1139.
- Heba Bou KaedBey, Mark van Hoeij, and Man Cheung Tsui, Solving Third Order Linear Difference Equations in Terms of Second Order Equations, arXiv:2402.11121 [math.AC], 2024. See p. 3.
Programs
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Maple
# A260772 satisfies a 4th-order recurrence that can be reduced # to a 2nd-order recurrence given in this program t: t := proc(n) options remember; if n <= 1 then [-1/2, 0, 1, 4][2*n+2] else (16*(n-2)*(2*n-3)*(5*n-2)*t(n-2) + (440*n^3-1056*n^2+724*n-144)*t(n-1)) /( n*(2*n+1)*(5*n-7) ) fi end: A260772 := proc(n) t(n/2) + ( (2-2*n)*t((n-1)/2)+(n+2)*t((n+1)/2) ) / (1+5*n) end: seq(A260772(i),i=0..100); # Mark van Hoeij, Jul 14 2022
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Maxima
a(n):=if n=0 then 1 else sum((-1)^j*binomial(n,j)*binomial(3*n-4*j,n-4*j+1),j,0,(n+1)/4)/n; /* Vladimir Kruchinin, Apr 04 2019 */
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PARI
a(n) = if (n==0, 1, sum(j=0, (n+1)/4, (-1)^j*binomial(n,j)*binomial(3*n-4*j, n-4*j+1))/n); \\ Michel Marcus, Apr 05 2019
Formula
G.f.: P1(x) = (2*(1-x)/3)/x - ((2*sqrt(1-5*x-2*x^2)/3)/x)*sin((Pi/6 + arccos(((20*x^3-6*x^2+15*x-2)/2)/(1-5*x-2*x^2)^(3/2))/3)). - See Dziemianczuk (2014), Proposition 11.
a(n) = (1/n)*Sum_{j=0..(n+1)/4} (-1)^j*C(n,j)*C(3*n-4*j,n-4*j+1), a(0)=1. - Vladimir Kruchinin, Apr 04 2019
n*(n+1)*(25*n^2-70*n+21)*a(n) - 30*(7*n-15)*n*a(n-1) + (-1100*n^4+5280*n^3-6424*n^2-1188*n+3816)*a(n-2) + 120*(n+2)*(n-3)*a(n-3) - 16*(n-3)*(n-4)*(25*n^2-20*n-24)*a(n-4) = 0. - Mark van Hoeij, Jul 14 2022
a(n) ~ 2^(n - 1/2) * phi^((10*n - 1)/4) / (sqrt(Pi) * 5^(1/4) * sqrt(phi^(3/2) - 2) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jul 15 2022
Extensions
More terms from Lars Blomberg, Aug 01 2015
Comments