This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A326792 #9 Jul 29 2019 12:00:40
%S A326792 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,
%T A326792 13,61,136,155,86,19,1,1,15,85,240,371,314,135,24,1,1,17,113,388,763,
%U A326792 882,585,202,29,1,1,19,145,588,1411,2086,1899,1019,290,35,1
%N 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.
%C A326792 Equivalent definition: T(n,k) equals the number of triangle stacks, as defined in A224704, containing n up-triangles and k down-triangles.
%C A326792 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.
%C A326792 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.
%H A326792 P. Bala, <a href="/A326792/a326792.pdf">Illustration for terms of row 4</a>
%H A326792 P. Bala, <a href="/A326792/a326792_1.pdf">Notes on A326792</a>
%F A326792 O.g.f. as a continued fraction including initial term 1: (u marks up-triangles and d marks down-triangles)
%F A326792 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 + ....
%F A326792 A(u,d) = 1/(2 - (1 + u)/(2 - (1 + u^2*d)/(2 - (1 + u^3*d^2)/(2 - (...) )))).
%F A326792 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 )
%F A326792 Row sums = A326793.
%e A326792 Triangle begins
%e A326792 n\k| 0 1 2 3 4 5 6 7 8 9
%e A326792 - - - - - - - - - - - - - - - - - - - - -
%e A326792 1 | 1
%e A326792 2 | 1 1
%e A326792 3 | 1 3 1
%e A326792 4 | 1 5 5 1
%e A326792 5 | 1 7 13 8 1
%e A326792 6 | 1 9 25 28 11 1
%e A326792 7 | 1 11 41 68 51 15 1
%e A326792 8 | 1 13 61 136 155 86 19 1
%e A326792 9 | 1 15 85 240 371 314 135 24 1
%e A326792 10 | 1 17 113 388 763 882 585 202 29 1
%e A326792 ...
%Y A326792 Row sums A326793. Cf. A224704.
%K A326792 nonn,tabl,easy
%O A326792 1,5
%A A326792 _Peter Bala_, Jul 25 2019