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
Keywords
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
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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
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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
Comments