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A326454
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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.
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5
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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
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0,7
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COMMENTS
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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 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.
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.
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LINKS
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FORMULA
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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).
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 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.
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EXAMPLE
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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
...
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CROSSREFS
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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STATUS
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approved
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