A326792 Triangular array: T(n,k) equals the number of small Schröder paths such that the area between the path and the x-axis contains n up-triangles and k down-triangles; n >= 1, k >= 0.

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 13, 8, 1, 1, 9, 25, 28, 11, 1, 1, 11, 41, 68, 51, 15, 1, 1, 13, 61, 136, 155, 86, 19, 1, 1, 15, 85, 240, 371, 314, 135, 24, 1, 1, 17, 113, 388, 763, 882, 585, 202, 29, 1, 1, 19, 145, 588, 1411, 2086, 1899, 1019, 290, 35, 1

Offset: 1

Author

Peter Bala, Jul 25 2019

Keywords

• nonn
• tabl
• easy

Comments

Equivalent definition: T(n,k) equals the number of triangle stacks, as defined in A224704, containing n up-triangles and k down-triangles.

We define two types of plane triangles - up-triangles with vertices at the integer lattice points (x, y), (x+1, y+1) and (x+2, y) and down-triangles with vertices at the integer lattice points (x, y), (x-1, y+1) and (x+1, y+1). The area beneath a small Schröder path may be decomposed in a unique manner into a collection of up- and down-triangles.

To construct a triangle stack (of small Schröder type) we start with a horizontal row of k contiguous up-triangles forming the base row of the stack. Subsequent rows of the stack are formed by placing down-triangles in some, all or none of the spaces between the up-triangles of the previous row. Further up-triangles may be then be placed on these down-triangles and the process repeated. For an example, see the illustration in the Links section. There is an obvious bijective correspondence between triangle stacks with a base of m up-triangles and small Schröder paths of semilength m.

Examples

			Triangle begins
  n\k| 0   1   2   3   4   5   6   7  8   9
  - - - - - - - - - - - - - - - - - - - - -
   1 | 1
   2 | 1   1
   3 | 1   3   1
   4 | 1   5   5   1
   5 | 1   7  13   8   1
   6 | 1   9  25  28  11   1
   7 | 1  11  41  68  51  15   1
   8 | 1  13  61 136 155  86  19   1
   9 | 1  15  85 240 371 314 135  24  1
  10 | 1  17 113 388 763 882 585 202 29  1
  ...
		

Links

• P. Bala, Illustration for terms of row 4
• P. Bala, Notes on A326792

Crossrefs

Row sums A326793. Cf. A224704.

Formula

O.g.f. as a continued fraction including initial term 1: (u marks up-triangles and d marks down-triangles)

A(u,d) = 1/(1 - u/(1 - u*d - u^2*d/(1 - u^2*d^2 - u^3*d^2/(1 - u^3*d^3 - u^4*d^3/(1 - u^4*d^4 - (...) ))))) = 1 + u + (1 + d)*u^2 + (1 + 3*d + d^2)*u^3 + ....

A(u,d) = 1/(2 - (1 + u)/(2 - (1 + u^2*d)/(2 - (1 + u^3*d^2)/(2 - (...) )))).

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^2+n)*d^(n^2)/( Product_{k = 1..n} ( 1 - (u*d)^k )^2 ) and D(u,d) = Sum_{n >= 0} (-1)^n*u^(n^2)*d^(n^2-n)/( Product_{k = 1..n} ( 1 - (u*d)^k )^2 )

Row sums = A326793.