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A007785 Number of sets of positive integers <= n^2 whose sum is (n^3 + n)/2.

%I A007785 #28 May 20 2022 05:17:20
%S A007785 1,1,2,17,306,10828,654857,63019177,9183937890,1953896126383,
%T A007785 589909767142505,247074213707554144,140902072248206260266,
%U A007785 107704589610917073318533,108877374411946899963718973,143864444783939220165210185294,245934054410000090878614435736720
%N A007785 Number of sets of positive integers <= n^2 whose sum is (n^3 + n)/2.
%H A007785 Alois P. Heinz, <a href="/A007785/b007785.txt">Table of n, a(n) for n = 0..40</a> (terms n = 1..18 from Sean A. Irvine)
%e A007785 a(2) = 2: {1,4}, {2,3}.
%e A007785 a(3) = 17: {6,9}, {7,8}, {1,5,9}, {1,6,8}, {2,4,9}, {2,5,8}, {2,6,7}, {3,4,8}, {3,5,7}, {4,5,6}, {1,2,3,9}, {1,2,4,8}, {1,2,5,7}, {1,3,4,7}, {1,3,5,6}, {2,3,4,6}, {1,2,3,4,5}.
%p A007785 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i*(i+1)/2<n, 0,
%p A007785       b(n, i-1) +`if`(i>n, 0, b(n-i, min(n-i, i-1)))))
%p A007785     end:
%p A007785 a:= n-> (s-> b(n*(1+s)/2, s))(n^2):
%p A007785 seq(a(n), n=0..16);  # _Alois P. Heinz_, Nov 02 2018
%t A007785 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i*(i + 1)/2 < n, 0, b[n, i - 1] + If[i > n, 0, b[n - i, Min[n - i, i - 1]]]]];
%t A007785 a[n_] := With[{s = n^2}, b[n*(1 + s)/2, s]];
%t A007785 Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, May 20 2022, after _Alois P. Heinz_ *)
%Y A007785 Cf. A052456.
%K A007785 nonn
%O A007785 0,3
%A A007785 Hidetoshi MINO [ mino(AT)hep.esb.yamanashi.ac.jp, mino(AT)mino.scri.fsu.edu ]
%E A007785 Corrected and extended by _David W. Wilson_
%E A007785 a(12) corrected and more terms from _Sean A. Irvine_, Jan 27 2018
%E A007785 a(0)=1 prepended by _Alois P. Heinz_, Nov 02 2018