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A156700 Number of partitions of the set of odd numbers {1, 3, 5, ..., 4*n-1} into two subsets with equal sum. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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

LINKS

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

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

EXAMPLE

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)

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 *)

PROG

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

CROSSREFS

Cf. A290889.

Sequence in context: A005630 A100507 A223006 * A274479 A231524 A182645

Adjacent sequences:  A156697 A156698 A156699 * A156701 A156702 A156703

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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Last modified May 17 12:55 EDT 2021. Contains 343971 sequences. (Running on oeis4.)