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%I #13 Jul 19 2019 15:11:22
%S 1,1,1,2,1,4,4,2,1,6,12,12,8,4,2,1,8,24,38,40,32,24,16,8,4,2,1,10,40,
%T 88,128,140,130,112,88,64,44,28,16,8,4,2,1,12,60,170,3320,448,512,520,
%U 488,428,358,288,220,160,112,76,48,28,16,8,4,2
%N Irregular triangle read by rows: T(n,k) is the number of small Schröder paths of semilength n such that the area between the path and the x-axis contains k down-triangles.
%C A Schröder path is a lattice path in the plane starting and ending on the x-axis, never going below the x-axis, using the steps (1,1) rise, (1,-1) fall or (2,0) flat. A small Schröder path is a Schröder path with no flat steps on the x-axis.
%C The area between a small Schröder path and the x-axis may be decomposed into a stack of unit area triangles; the triangles come in two types: up-triangles with vertices at the lattice points (x, y), (x+1, y+1) and (x+2, y) and down-triangles with vertices at the lattice points (x, y), (x-1, y+1) and (x+1, y+1). These are the triangle stacks of A224704. Here we enumerate triangle stacks with n >= 1 up-triangles in the bottom row of the stack (corresponding to small Schröder paths of semilength n) and containing k >= 0 down-triangles in the stack. See the illustration in the Links section for an example.
%H P. Bala, <a href="/A224704/a224704.pdf">The area beneath small Schröder paths: Notes on A224704, A326453 and A326454</a>
%H P. Bala, <a href="/A309086/a309086.pdf">Illustration for row 3</a>
%F O.g.f. as a continued fraction: A(u,d) = 1/(1 - u/(1 - u*d - u*d/(1 - u*d^2 - u*d^2/(1 - u*d^3 - (...) )))) = 1 + u + (1 + 2*d)*u^2 + (1 + 4*d + 4*d^2 + 2*d^3)*u^3 + ... (u marks the semilength of the path (or, equivalently, up-triangles in the bottom row of the associated triangle stack) and d marks down-triangles in the stack).
%F Other continued fractions: A(u,d) = 1/(1 + u - 2*u/(1 + u - (1 + d)*u/(1 + u - (1 + d^2)*u/(1 + u - (...) )))).
%F A(u,d) = 1/(1 - u/(1 - (d + d)*u/(1 - d^2*u/(1 - (d^2 + d^3)*u/(1 - d^4*u/(1 - (d^3 + d^5)*u/(1 - d^6*u/(1 - (d^4 + d^7)*u/(1 - (...) ))))))))).
%F O.g.f. as a ratio of q-series: N(u,d)/D(u,d), where N(u,d) = Sum_{n >= 0} (-1)^n*u^n*d^(n^2)/( (1 - d)*(1 - d^2)*...*(1 - d^n) * (1 - u*d)*(1 - u*d^2)*...*(1 - u*d^n) ) and D(u,d) = Sum_{n >= 0} (-1)^n*u^n*d^(n(n-1))/( (1 - d)*(1 - d^2)*...*(1 - d^n) * (1 - u*d)*(1 - u*d^2)*...*(1 - u*d^n) ).
%e n\k | 0 1 2 3 4 5 6 7 8 9 10
%e - - - - - - - - - - - - - - - - - - - - - - - - - - - -
%e 0 | 1
%e 1 | 1
%e 2 | 1 2
%e 3 | 1 4 4 2
%e 4 | 1 6 12 12 8 4 2
%e 5 | 1 8 24 38 40 32 24 16 8 4 2
%e ...
%Y Row sums A001003. Cf. A224704, A227543, A326453, A326454.
%K nonn,tabf,easy
%O 0,4
%A _Peter Bala_, Jul 16 2019
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