

A326792


Triangular array: T(n,k) equals the number of small Schröder paths such that the area between the path and the xaxis contains n uptriangles and k downtriangles; n >= 1, k >= 0.


2



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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,5


COMMENTS

Equivalent definition: T(n,k) equals the number of triangle stacks, as defined in A224704, containing n uptriangles and k downtriangles.
We define two types of plane triangles  uptriangles with vertices at the integer lattice points (x, y), (x+1, y+1) and (x+2, y) and downtriangles with vertices at the integer lattice points (x, y), (x1, 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 downtriangles.
To construct a triangle stack (of small Schröder type) we start with a horizontal row of k contiguous uptriangles forming the base row of the stack. Subsequent rows of the stack are formed by placing downtriangles in some, all or none of the spaces between the uptriangles of the previous row. Further uptriangles may be then be placed on these downtriangles 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 uptriangles and small Schröder paths of semilength m.


LINKS



FORMULA

O.g.f. as a continued fraction including initial term 1: (u marks uptriangles and d marks downtriangles)
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 qseries: 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^2n)/( Product_{k = 1..n} ( 1  (u*d)^k )^2 )


EXAMPLE

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
...


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



