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A326793
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The number of small Schröder paths such that the area between the path and the x-axis contains n up-triangles.
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2
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1, 1, 2, 5, 12, 30, 75, 188, 472, 1186, 2981, 7494, 18842, 47376, 119126, 299545, 753220, 1894018, 4762640, 11976010, 30114592, 75725485, 190417684, 478820320, 1204031670, 3027633300, 7613224740, 19144059492, 48139261637, 121050006438
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OFFSET
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0,3
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COMMENTS
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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. This decomposition produces a triangle stack in the sense of A224704. Here we are counting triangle stacks containing n up-triangles. See the Links section for an illustration.
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LINKS
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FORMULA
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O.g.f. as a continued fraction: (u marks up-triangles)
A(u) = 1/(1 - u/(1 - u - u^2/(1 - u^2 - u^3/(1 - u^3 - u^4/(1 - u^4 - (...) ))))) = 1 + u + 2*u^2 + 5*u^3 + 12*u^4 + ....
A(u) = 1/(1 - u/(1 - (u + u^2)/(1 - u^3/(1 - (u^2 + u^4)/(1 - u^5/(1 - (u^3 + u^6)/(1 - u^7/( (...) )))))))).
A(u) = 1/(2 - (1 + u)/(2 - (1 + u^2)/(2 - (1 + u^3)/(2 - (...) )))).
A(u) = N(u)/D(u), where N(u) = Sum_{n >= 0} u^(n^2+n)/ Product_{k = 1..n} ((1 - u^k)^2) and D(u) = Sum_{n >= 0} u^(n^2)/ Product_{k = 1..n} ((1 - u^k)^2).
a(n) ~ c*d^n, where c = 0.29475 98606 22204 98206 41002 ..., d = 2.51457 96438 78729 18851 04371 ....
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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