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%I #42 May 31 2020 22:21:49
%S 1,1,1,3,7,18,51,155,486,1555,5095,17038,57801,198471,689039,2415043,
%T 8534022,30375188,108815273,392076629,1420064031,5167575997,
%U 18885299641,69287981666,255121926519,942474271999,3492314839349,12977225566680,48349025154154
%N Number of n-element subsets that can be chosen from {1,2,...,2*n} having element sum n^2.
%C a(n) is the number of partitions of n^2 into n distinct parts <= 2*n.
%C 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
%C 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
%H Alois P. Heinz, <a href="/A202261/b202261.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) ~ sqrt(3) * 4^n / (Pi * n^2). - _Vaclav Kotesovec_, Sep 10 2014
%e a(0) = 1: {}.
%e a(1) = 1: {1}.
%e a(2) = 1: {1,3}.
%e a(3) = 3: {1,2,6}, {1,3,5}, {2,3,4}.
%e 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}.
%p b:= proc(n, i, t) option remember;
%p `if`(i<t or n<t*(t+1)/2 or n>t*(2*i-t+1)/2, 0,
%p `if`(n=0, 1, b(n, i-1, t) +`if`(n<i, 0, b(n-i, i-1, t-1))))
%p end:
%p a:= n-> b(n^2, 2*n, n):
%p seq(a(n), n=0..30);
%t 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_ *)
%Y Column k=1 of A185282.
%Y Cf. A000217, A000290, A107379, A204459.
%K nonn
%O 0,4
%A _Alois P. Heinz_, Jan 20 2012
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