A232534 Number of subsets of {1,...,n} containing n and having at least one set partition into 3 blocks with equal element sum.

0, 0, 0, 0, 1, 2, 5, 12, 29, 63, 146, 329, 722, 1613, 3505, 7567, 16119, 34194, 71455, 148917, 307432, 631816, 1290905, 2628736, 5330368

Offset: 1

Author

Alois P. Heinz, Nov 25 2013

Keywords

• nonn
• more

Comments

Subsets with more than one set partition into 3 blocks with equal element sum are counted only once: {1,2,3,4,5,6,7,8}-> 1236/48/57, 138/246/57, 156/237/48.

Examples

			a(5) = 1: {1,2,3,4,5}-> 14/23/5.
a(6) = 2: {1,2,4,5,6}-> 15/24/6, {1,2,3,4,5,6}-> 16/25/34.
a(7) = 5: {2,3,4,5,7}-> 25/34/7, {1,3,4,6,7}-> 16/34/7, {1,2,5,6,7}-> 16/25/7, {1,2,3,5,6,7}-> 17/26/35, {2,3,4,5,6,7}-> 27/36/45.
a(8) = 12: {2,3,5,6,8}, {1,3,5,7,8}, {1,2,6,7,8}, {2,3,4,6,7,8}, {1,2,3,4,5,7,8}, {1,3,4,5,6,8}, {1,2,4,5,6,7,8}, {1,2,3,6,7,8}, {3,4,5,6,7,8}, {1,2,4,5,7,8}, {1,2,3,4,5,6,7,8}, {1,2,3,4,6,8}.
		

Crossrefs

Cf. A164934, A232466 (2 blocks).

Column k=3 of A248112.

Programs

Maple

b:= proc(n, k, i) option remember; local m; m:= i*(i+1)/2;
      `if`(k>n, b(k, n, i), `if`(i<1, `if`(n=0 and k=0, {0}, {}),
      `if`(k>=0 and n+k>m or k<0 and n-2*k>m, {}, b(n, k, i-1)
       union map(p-> p+x^i, b(n+i, k+i, i-1) union b(n-i, k, i-1)
       union b(n, k-i, i-1)))))
    end:
a:= n-> nops(b(n, n, n-1)):
seq(a(n), n=1..15);

Mathematica

b[n_, k_, i_] := b[n, k, i] = Module[{m = i*(i + 1)/2}, If[k > n, b[k, n, i], If[i < 1, If[n == 0 && k == 0, {0}, {}], If[k >= 0 && n + k > m || k < 0 && n - 2*k > m, {}, b[n, k, i - 1] ~Union~ Map[# + x^i &, b[n + i, k + i, i - 1] ~Union~ b[n - i, k, i - 1] ~Union~ b[n, k - i, i - 1]]]]]];
a[n_] := Length[b[n, n, n - 1]];
Table[a[n], {n, 1, 20}] (* Jean-François Alcover, May 25 2018, translated from Maple *)

Extensions

a(25) from Alois P. Heinz, Mar 26 2016