A326454 - OEIS

%I #16 Jul 19 2019 15:10:46

%S 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,

%T 11,1,17,85,136,51,1,1,19,113,240,155,15,1,21,145,388,371,86,1,1,23,

%U 181,588,763,314,19

%N Irregular triangle read by rows: T(n,k) is the number of small Schröder paths such that the area between the path and the x-axis is equal to n and contains k down-triangles.

%C A227543 is the companion triangle for Dyck paths.

%C Number of n triangle stacks, in the sense of A224704, containing k down- triangles.

%C 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.

%C 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). See the illustration in the Links section for an example.

%H P. Bala, <a href="/A326454/a326454.pdf">Illustration for row 5</a>

%H P. Bala, <a href="/A224704/a224704.pdf">The area beneath small Schröder paths: Notes on A224704, A326453 and A326454</a>

%F 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 down-triangles).

%F 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 - (...) )))).

%F 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/( (...) )))))))).

%F O.g.f. as a ratio of q-series: 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.

%e Triangle begins

%e   n\k|  0    1    2     3    4

%e ------------------------------

%e    0 |  1

%e    1 |  1

%e    2 |  1

%e    3 |  1    1

%e    4 |  1    3

%e    5 |  1    5    1

%e    6 |  1    7    5

%e    7 |  1    9   13    1

%e    8 |  1   11   25    8

%e    9 |  1   13   41   28    1

%e   10 |  1   15   61   68   11

%e   ...

%Y Row sums A224704. Cf. A001003, A227543, A309086, A326453.

%K nonn,tabf,easy

%O 0,7

%A _Peter Bala_, Jul 06 2019