A260770 Certain directed lattice paths.

1, 6, 35, 207, 1251, 7678, 47658, 298371, 1880659, 11918586, 75871710, 484793950, 3107494430, 19973075580, 128678167220, 830735862179, 5372968238979, 34807369089378, 225818672567382, 1466956891774602, 9540909022501226, 62119854068610436, 404854330511525580

Offset: 0

Author

N. J. A. Sloane, Jul 30 2015

Keywords

• nonn

Comments

See Dziemianczuk (2014) for precise definition.

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.

Formula

See Dziemianczuk (2014) Equation (29a) with m=1.

From Vaclav Kotesovec, Jul 15 2022: (Start)

Recurrence: (n-2)*n*(n+1)*(100*n^3 - 510*n^2 + 677*n - 111)*a(n) = -6*n*(40*n^3 - 5*n^2 - 586*n + 863)*a(n-1) + 4*(n-1)*(1100*n^5 - 6710*n^4 + 12387*n^3 - 3775*n^2 - 8723*n + 5448)*a(n-2) - 72*(n-2)*(n-1)*(10*n^2 - 5*n - 24)*a(n-3) + 16*(n-3)*(n-2)*(n-1)*(100*n^3 - 210*n^2 - 43*n + 156)*a(n-4).

a(n) ~ sqrt((4*phi^6 - 1)/5 + phi^(11/2)) * 2^(n-1) * phi^(5*n/2) / sqrt(Pi*n), where phi = A001622 is the golden ratio. (End)

Extensions

More terms from Lars Blomberg, Aug 01 2015