

A326454


Irregular triangle read by rows: T(n,k) is the number of small Schröder paths such that the area between the path and the xaxis is equal to n and contains k downtriangles.


5



1, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 7, 5, 1, 9, 13, 1, 1, 11, 25, 8, 1, 13, 41, 28, 1, 1, 15, 61, 68, 11, 1, 17, 85, 136, 51, 1, 1, 19, 113, 240, 155, 15, 1, 21, 145, 388, 371, 86, 1, 1, 23, 181, 588, 763, 314, 19
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OFFSET

0,7


COMMENTS

A227543 is the companion triangle for Dyck paths.
Number of n triangle stacks, in the sense of A224704, containing k down triangles.
A Schröder path is a lattice path in the plane starting and ending on the xaxis, never going below the xaxis, 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 xaxis.
The area between a small Schröder path and the xaxis may be decomposed into a stack of unit area triangles; the triangles come in two types: uptriangles with vertices at the lattice points (x, y), (x+1, y+1) and (x+2, y) and downtriangles with vertices at the lattice points (x, y), (x1, y+1) and (x+1, y+1). See the illustration in the Links section for an example.


LINKS



FORMULA

O.g.f. as a continued fraction: A(q,d) = 1/(2  (1 + q)/(2  (1 + q^3*d)/(2  (1 + q^5*d^2)/( (...) )))) = 1 + q + q^2 + q^3*(1 + d) + q^4*(1 + 3*d) + q^5*(1 + 5*d + d^2) + ... (q marks the area, d marks downtriangles).
Other continued fractions: A(q,d) = 1/(1  q/(1  q^2*d  q^3*d/(1  q^4*d^2  q^5*d^2/(1  q^6*d^3  (...) )))).
A(q,d) = 1/(1  q/(1  (q^2*d + q^3*d)/(1  q^5*d^2/(1  (q^4*d^2 + q^7*d^3)/(1  q^9*d^4/(1  (q^6*d^3 + q^11*d^5)/(1  q^13*d^6/( (...) )))))))).
O.g.f. as a ratio of qseries: N(q,d)/D(q,d), where N(q,d) = Sum_{n >= 0} (1)^n*d^(n^2)*q^(2*n^2 + n)/( (1  d*q^2)*(1  d^2*q^4)*...*(1  d^n*q^(2*n)) )^2 and D(q,d) = Sum_{n >= 0} (1)^n*d^(n^2  n)*q^(2*n^2  n)/( (1  d*q^2)*(1  d^2*q^4)*...*(1  d^n*q^(2*n)) )^2.


EXAMPLE

Triangle begins
n\k 0 1 2 3 4

0  1
1  1
2  1
3  1 1
4  1 3
5  1 5 1
6  1 7 5
7  1 9 13 1
8  1 11 25 8
9  1 13 41 28 1
10  1 15 61 68 11
...


CROSSREFS



KEYWORD

nonn,tabf,easy


AUTHOR



STATUS

approved



