A202261 Number of n-element subsets that can be chosen from {1,2,...,2*n} having element sum n^2.

1, 1, 1, 3, 7, 18, 51, 155, 486, 1555, 5095, 17038, 57801, 198471, 689039, 2415043, 8534022, 30375188, 108815273, 392076629, 1420064031, 5167575997, 18885299641, 69287981666, 255121926519, 942474271999, 3492314839349, 12977225566680, 48349025154154

Offset

0,4

Comments

a(n) is the number of partitions of n^2 into n distinct parts <= 2*n.

Taking the complement of each set, a(n) is also the number of partitions of n^2+n into n distinct parts <= 2*n. - Franklin T. Adams-Watters, Jan 20 2012

Also the number of partitions of n*(n+1)/2 into at most n parts of size at most n. a(4) = 7: 433, 442, 3322, 3331, 4222, 4321, 4411. - Alois P. Heinz, May 31 2020

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Formula

a(n) ~ sqrt(3) * 4^n / (Pi * n^2). - Vaclav Kotesovec, Sep 10 2014

Example

a(0) = 1: {}.
a(1) = 1: {1}.
a(2) = 1: {1,3}.
a(3) = 3: {1,2,6}, {1,3,5}, {2,3,4}.
a(4) = 7: {1,2,5,8}, {1,2,6,7}, {1,3,4,8}, {1,3,5,7}, {1,4,5,6}, {2,3,4,7},{2,3,5,6}.

Maple

b:= proc(n, i, t) option remember;
      `if`(i<t or n<t*(t+1)/2 or n>t*(2*i-t+1)/2, 0,
      `if`(n=0, 1, b(n, i-1, t) +`if`(n<i, 0, b(n-i, i-1, t-1))))
    end:
a:= n-> b(n^2, 2*n, n):
seq(a(n), n=0..30);

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[i<t || n<t*(t+1)/2 || n>t*(2*i-t+1)/2, 0, If[n == 0, 1, b[n, i-1, t] + If[n<i, 0, b[n-i, i-1, t-1]]]]; a[n_] = b[n^2, 2*n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 05 2015, after Alois P. Heinz *)

Crossrefs

Column k=1 of A185282.

Cf. A000217, A000290, A107379, A204459.

Sequence in context: A218783 A103177 A262784 * A180166 A369842 A256873

Adjacent sequences: A202258 A202259 A202260 * A202262 A202263 A202264

Keyword

nonn

Author

Alois P. Heinz, Jan 20 2012

Status

approved