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A326453 Triangle read by rows: T(n,k) is the number of small Schröder paths of semilength k such that the area between the path and the x-axis is equal to n (n >= 0; 0 <= k <= n). 5
1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 0, 3, 3, 1, 0, 0, 0, 2, 6, 4, 1, 0, 0, 0, 1, 7, 10, 5, 1, 0, 0, 0, 1, 6, 16, 15, 6, 1, 0, 0, 0, 1, 5, 19, 30, 21, 7, 1, 0, 0, 0, 0, 5, 19, 45, 50, 28, 8, 1, 0, 0, 0, 0, 4, 19, 55, 90, 77, 36, 9, 1, 3, 19, 61, 131, 161, 112, 45, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,14

COMMENTS

A239927 is the companion triangle for Dyck paths.

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 are of 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). A small Schröder path of semilength k has k up-triangles in the bottom row of its stack. See the illustration in the Links section for an example. Thus an alternative description of the triangle entry T(n,k) is the number of n triangle stacks, in the sense of A224704, containing k up-triangles in the bottom row.

LINKS

Table of n, a(n) for n=0..86.

P. Bala, Illustration for row 5

P. Bala, The area beneath small Schröder paths: Notes on A224704, A326453 and A326454

FORMULA

O.g.f. as a continued fraction: A(q,u) = 1/(1 + u - (1 + q)*u/(1 + u - (1 + q^3)*u/(1 + u - (1 + q^5)*u/( (...) )))) =  1 + q*u + q^2*u^2 + q^3*(u^2 + u^3) + q^4*(u^2 + 2*u^3 + u^4) + ...(q marks the area, u marks the up- triangles in the bottom row).

Alternative forms: A(q,u) = 1/(1 - q*u/(1 - q^2*u - q^3*u/(1 - q^4*u/( (...) ))));

A(q,u) = 1/(1 - q*u/(1 - (q^2 + q^3)*u/(1 - q^5*u/(1 - (q^4 + q^7)*u/(1 - q^9*u/(1 - (q^6 + q^11)*u/(1 - q^13*u/( (...) )))))))).

O.g.f. as a ratio of q-series: N(q,u)/D(q,u), where N(q,u) = Sum_{n >= 0} (-1)^n*u^n*q^(2*n^2 + n)/( (1 - q^2)*(1 - q^4)*...*(1 - q^(2*n)) * (1 - u*q^2)*(1 - u*q^4)*...*(1 - u*q^(2*n)) ) and D(q,u) = Sum_{n >= 0} (-1)^n*u^n*q^(2*n^2 - n)/( (1 - q^2)*(1 - q^4)*...*(1 - q^(2*n)) * (1 - u*q^2)*(1 - u*q^4)*...*(1 - u*q^(2*n)) ).

EXAMPLE

Triangle begins

  n\k|  0    1   2    3    4    5    6    7   8    9

  --------------------------------------------------

   0 |  1

   1 |  0    1

   2 |  0    0   1

   3 |  0    0   1    1

   4 |  0    0   1    2    1

   5 |  0    0   0    3    3    1

   6 |  0    0   0    2    6    4    1

   7 |  0    0   0    1    7   10    5    1

   8 |  0    0   0    1    6   16   15    6   1

   9 |  0    0   0    1    5   19   30   21   7   1

   ...

Example of a stack of 10 up- and down-triangles with 5 up-triangles in the bottom row.

          /\  /\

         /__\/__\     __

        /\  /\  /\  /\  /\

       /__\/__\/__\/__\/__\

CROSSREFS

Row sums A224704. Cf. A047998, A227543, A239927, A309086, A326454.

Sequence in context: A036869 A145466 A036868 * A130116 A325774 A350488

Adjacent sequences:  A326450 A326451 A326452 * A326454 A326455 A326456

KEYWORD

nonn,tabl,easy

AUTHOR

Peter Bala, Jul 06 2019

STATUS

approved

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Last modified October 3 22:17 EDT 2022. Contains 357237 sequences. (Running on oeis4.)