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A156700
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Number of partitions of the set of odd numbers {1, 3, 5, ..., 4*n-1} into two subsets with equal sum.
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9
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0, 1, 1, 4, 10, 34, 103, 346, 1153, 3965, 13746, 48396, 171835, 615966, 2223755, 8082457, 29543309, 108545916, 400623807, 1484716135, 5522723344, 20612084010, 77164686511, 289688970195, 1090342139349, 4113620233260, 15553877949800, 58930127470164
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OFFSET
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1,4
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COMMENTS
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Also the number of 2 X 2n reduced magic rectangles with values 1..4n. In a magic rectangle all column sums are equal and also all row sums are equal. Reduced means up to row and column permutations. - Andrew Howroyd, Nov 22 2018
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LINKS
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FORMULA
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a(n) ~ sqrt(3) * 2^(2*n-3) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 18 2017
a(n) = [x^0](Product_{k=1..2*n} x^-(2*k-1) + x^(2*k-1))/2. - Andrew Howroyd, Nov 22 2018
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EXAMPLE
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For n=2: {1,7}U{3,5}. For n=3: {1,3,5,9}U{7,11}. For n=4: {1,3,13,15}U{5,7,9,11}, {1,5,11,15}U{3,7,9,13}, {1,7,9,15}U{3,5,11,13}, {3,5,9,15}U{1,7,11,13}.
For n=3: The unique 2 X 6 reduced magic rectangle is:
1 3 7 8 9 11
12 10 6 5 4 2
(End)
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MAPLE
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b:= proc() option remember; local i, j, t; `if`(args[1]=0, `if`(nargs=2, 1, b(args[t] $t=2..nargs)), add(`if`(args[j] -args[nargs] <0, 0, b(sort([seq(args[i] -`if`(i=j, args[nargs], 0), i=1..nargs-1)])[], args[nargs]-2)), j=1..nargs-1)) end: a:= n-> b((2*n^2)$2, 4*n-1)/2: seq(a(n), n=1..40); # Alois P. Heinz, Sep 06 2009
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MATHEMATICA
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Table[SeriesCoefficient[Product[(x^(2*k - 1) + 1/x^(2*k - 1)), {k, 1, 2*n}]/2, {x, 0, 0}], {n, 1, 30}] (* G. C. Greubel, Nov 22 2018 *)
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PROG
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(PARI) a(n)=polcoef(prod(k=1, 2*n, x^-(2*k-1) + x^(2*k-1)), 0)/2; \\ Andrew Howroyd, Nov 22 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Wim Couwenberg (wim.couwenberg(AT)gmail.com), Feb 13 2009
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EXTENSIONS
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STATUS
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approved
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