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

1, 1, 2, 17, 306, 10828, 654857, 63019177, 9183937890, 1953896126383, 589909767142505, 247074213707554144, 140902072248206260266, 107704589610917073318533, 108877374411946899963718973, 143864444783939220165210185294, 245934054410000090878614435736720

Offset: 0

Author

Hidetoshi MINO [ mino(AT)hep.esb.yamanashi.ac.jp, mino(AT)mino.scri.fsu.edu ]

Keywords

• nonn

Examples

			a(2) = 2: {1,4}, {2,3}.
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}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..40 (terms n = 1..18 from Sean A. Irvine)

Crossrefs

Cf. A052456.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i*(i+1)/2n, 0, b(n-i, min(n-i, i-1)))))
    end:
a:= n-> (s-> b(n*(1+s)/2, s))(n^2):
seq(a(n), n=0..16);  # Alois P. Heinz, Nov 02 2018

Mathematica

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]]]]];
a[n_] := With[{s = n^2}, b[n*(1 + s)/2, s]];
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, May 20 2022, after Alois P. Heinz *)

Extensions

Corrected and extended by David W. Wilson

a(12) corrected and more terms from Sean A. Irvine, Jan 27 2018

a(0)=1 prepended by Alois P. Heinz, Nov 02 2018