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%I #36 Nov 22 2018 15:00:37
%S 0,1,1,4,10,34,103,346,1153,3965,13746,48396,171835,615966,2223755,
%T 8082457,29543309,108545916,400623807,1484716135,5522723344,
%U 20612084010,77164686511,289688970195,1090342139349,4113620233260,15553877949800,58930127470164
%N Number of partitions of the set of odd numbers {1, 3, 5, ..., 4*n-1} into two subsets with equal sum.
%C 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
%H Seiichi Manyama, <a href="/A156700/b156700.txt">Table of n, a(n) for n = 1..1000</a> (terms 1..400 from Alois P. Heinz)
%F a(n) ~ sqrt(3) * 2^(2*n-3) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Sep 18 2017
%F a(n) = [x^0](Product_{k=1..2*n} x^-(2*k-1) + x^(2*k-1))/2. - _Andrew Howroyd_, Nov 22 2018
%e 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}.
%e From _Andrew Howroyd_, Nov 22 2018: (Start)
%e For n=3: The unique 2 X 6 reduced magic rectangle is:
%e 1 3 7 8 9 11
%e 12 10 6 5 4 2
%e (End)
%p 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
%t 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 *)
%o (PARI) a(n)=polcoef(prod(k=1, 2*n, x^-(2*k-1) + x^(2*k-1)), 0)/2; \\ _Andrew Howroyd_, Nov 22 2018
%Y Cf. A290889.
%K nonn
%O 1,4
%A Wim Couwenberg (wim.couwenberg(AT)gmail.com), Feb 13 2009
%E Extended beyond a(18) by _Alois P. Heinz_, Sep 06 2009
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