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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.

%I A302285 #17 Apr 12 2025 08:45:37
%S A302285 1,2,7,33,185,1170,8121,60846,486753,4125852,36846557,345205559,
%T A302285 3381126995,34524194712,366635359887,4041180951473,46149726728969,
%U A302285 545161967955668,6652026230285141,83730953689450825,1085924693069106823,14494802798426546660,198918641942013097723
%N 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.
%C A302285 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
%H A302285 Alois P. Heinz, <a href="/A302285/b302285.txt">Table of n, a(n) for n = 0..549</a>
%H A302285 Veronica Bitonti, Bishal Deb, and Alan D. Sokal, <a href="https://arxiv.org/abs/2412.10214">Thron-type continued fractions (T-fractions) for some classes of increasing trees</a>, arXiv:2412.10214 [math.CO], 2024. See p. 58.
%e A302285 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 + ...
%p A302285 b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
%p A302285      `if`(x=0, 1, b(x-1, y-1)+b(x-1, y+1)+b(x-2, y)*(y+1)))
%p A302285     end:
%p A302285 a:= n-> b(2*n, 0):
%p A302285 seq(a(n), n=0..22);  # _Alois P. Heinz_, Apr 12 2025
%t A302285 nmax = 22; CoefficientList[Series[1/(1 - x + ContinuedFractionK[-x, 1 - (k + 1) x, {k, 1, nmax}]), {x, 0, nmax}], x]
%Y A302285 Cf. A001515, A006318, A006351, A006789, A258173, A295289.
%K A302285 nonn
%O A302285 0,2
%A A302285 _Ilya Gutkovskiy_, Apr 04 2018