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A356367
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Number of plane partitions of n having exactly one row and one column, each of equal length.
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1
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1, 1, 1, 2, 2, 5, 6, 11, 16, 26, 36, 58, 81, 122, 172, 251, 350, 502, 692, 972, 1332, 1842, 2499, 3414, 4592, 6200, 8277, 11064, 14656, 19424, 25544, 33584, 43880, 57274, 74362, 96429, 124468, 160422, 205942, 263938, 337083, 429768
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OFFSET
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0,4
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COMMENTS
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The empty plane partition of 0 contributes an initial term equal to 1.
Also equal to the number of unimodal compositions of n+1 where the peak appears exactly once and the number of parts to the left of the peak is equal to the number of parts to the right of the peak.
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LINKS
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FORMULA
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G.f.: 1 + (1/Product_{n>=1}(1-x^n)^2)*Sum_{r,n>=0}(-1)^(n+r+1)*x^(n*(n+1)/2 + r*(r+1)/2 + 2*n*r)*(1-x^r).
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EXAMPLE
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For n = 7 the valid unimodal compositions of n+1=8 are (8), (1,6,1), (1,5,2), (2,5,1), (3,4,1), (1,4,3), (2,4,2), (1,1,4,1,1), (1,1,3,2,1), (1,2,3,1,1) and (1,1,1,2,1,1,1), and so a(7) = 11.
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MATHEMATICA
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nmax = 50; CoefficientList[Series[1 + 1/Product[(1 - x^k)^2, {k, 1, nmax}] * Sum[(-1)^(k + r + 1) * x^(k*(k + 1)/2 + r*(r + 1)/2 + 2*k*r)*(1 - x^r), {r, 0, nmax}, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 28 2023 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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