A302285 Expansion of 1/(1 - x - x/(1 - 2*x - x/(1 - 3*x - x/(1 - 4*x - x/(1 - 5*x - x/(1 - ...)))))), a continued fraction.

1, 2, 7, 33, 185, 1170, 8121, 60846, 486753, 4125852, 36846557, 345205559, 3381126995, 34524194712, 366635359887, 4041180951473, 46149726728969, 545161967955668, 6652026230285141, 83730953689450825, 1085924693069106823, 14494802798426546660, 198918641942013097723

Offset: 0

Author

Ilya Gutkovskiy, Apr 04 2018

Keywords

• nonn

Comments

a(n) is the number of paths from (0,0) to (2n,0) on or above the x-axis with steps U=(1,1), D=(1,-1), and L=(2,0), where the level steps L at height k have k+1 colors for all k>=0. - Alexander Burstein, Apr 10 2025

Examples

			G.f. A(x) = 1 + 2*x + 7*x^2 + 33*x^3 + 185*x^4 + 1170*x^5 + 8121*x^6 + 60846*x^7 + 486753*x^8 + ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..549
• Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.

Crossrefs

Cf. A001515, A006318, A006351, A006789, A258173, A295289.

Programs

Maple

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, b(x-1, y-1)+b(x-1, y+1)+b(x-2, y)*(y+1)))
    end:
a:= n-> b(2*n, 0):
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 12 2025

Mathematica

nmax = 22; CoefficientList[Series[1/(1 - x + ContinuedFractionK[-x, 1 - (k + 1) x, {k, 1, nmax}]), {x, 0, nmax}], x]