A156700 Number of partitions of the set of odd numbers {1, 3, 5, ..., 4*n-1} into two subsets with equal sum.

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

Offset: 1

Author

Wim Couwenberg (wim.couwenberg(AT)gmail.com), Feb 13 2009

Keywords

• nonn

Comments

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

Examples

			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}.
From _Andrew Howroyd_, Nov 22 2018: (Start)
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)
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000 (terms 1..400 from Alois P. Heinz)

Crossrefs

Cf. A290889.

Programs

Maple

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

Mathematica

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 *)

PARI

a(n)=polcoef(prod(k=1, 2*n, x^-(2*k-1) + x^(2*k-1)), 0)/2; \\ Andrew Howroyd, Nov 22 2018

Formula

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

Extensions

Extended beyond a(18) by Alois P. Heinz, Sep 06 2009