A309086 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.

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, 88, 128, 140, 130, 112, 88, 64, 44, 28, 16, 8, 4, 2, 1, 12, 60, 170, 3320, 448, 512, 520, 488, 428, 358, 288, 220, 160, 112, 76, 48, 28, 16, 8, 4, 2

Offset: 0

Author

Peter Bala, Jul 16 2019

Keywords

• nonn
• tabf
• easy

Comments

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.

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.

Examples

			   n\k |  0    1    2    3    4    5    6    7   8   9  10
   - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    0  |  1
    1  |  1
    2  |  1    2
    3  |  1    4    4    2
    4  |  1    6   12   12    8    4    2
    5  |  1    8   24   38   40   32   24   16   8   4   2
   ...
		

Links

• P. Bala, The area beneath small Schröder paths: Notes on A224704, A326453 and A326454
• P. Bala, Illustration for row 3

Crossrefs

Row sums A001003. Cf. A224704, A227543, A326453, A326454.

Formula

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).

Other continued fractions: A(u,d) = 1/(1 + u - 2*u/(1 + u - (1 + d)*u/(1 + u - (1 + d^2)*u/(1 + u - (...) )))).

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 - (...) ))))))))).

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) ).