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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 x-axis is equal to n and contains k down-triangles.

%I A326454 #16 Jul 19 2019 15:10:46
%S A326454 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 A326454 11,1,17,85,136,51,1,1,19,113,240,155,15,1,21,145,388,371,86,1,1,23,
%U A326454 181,588,763,314,19
%N 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 x-axis is equal to n and contains k down-triangles.
%C A326454 A227543 is the companion triangle for Dyck paths.
%C A326454 Number of n triangle stacks, in the sense of A224704, containing k down- triangles.
%C A326454 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 A326454 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 A326454 P. Bala, <a href="/A326454/a326454.pdf">Illustration for row 5</a>
%H A326454 P. Bala, <a href="/A224704/a224704.pdf">The area beneath small Schröder paths: Notes on A224704, A326453 and A326454</a>
%F A326454 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 A326454 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 A326454 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 A326454 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 A326454 Triangle begins
%e A326454   n\k|  0    1    2     3    4
%e A326454 ------------------------------
%e A326454    0 |  1
%e A326454    1 |  1
%e A326454    2 |  1
%e A326454    3 |  1    1
%e A326454    4 |  1    3
%e A326454    5 |  1    5    1
%e A326454    6 |  1    7    5
%e A326454    7 |  1    9   13    1
%e A326454    8 |  1   11   25    8
%e A326454    9 |  1   13   41   28    1
%e A326454   10 |  1   15   61   68   11
%e A326454   ...
%Y A326454 Row sums A224704. Cf. A001003, A227543, A309086, A326453.
%K A326454 nonn,tabf,easy
%O A326454 0,7
%A A326454 _Peter Bala_, Jul 06 2019