A164934 Number of different ways to select 3 disjoint subsets from {1..n} with equal element sum.

0, 0, 0, 0, 1, 3, 8, 22, 63, 157, 502, 1562, 4688, 15533, 50953, 165054, 562376, 1911007, 6467143, 22447463, 78021923, 271410289, 957082911, 3384587525, 11998851674, 42876440587, 153684701645, 552421854011, 1995875594696, 7231871165277, 26274832876337

Offset: 1

Author

Alois P. Heinz, Aug 31 2009

Keywords

• nonn

Comments

a(5) = 1, because {1,4}, {2,3}, {5} are disjoint subsets of {1..5} with element sum 5.

a(6) = 3: {1,4}, {2,3}, {5} have element sum 5, {1,5}, {2,4}, {6} have element sum 6, and {1,6}, {2,5}, {3,4} have element sum 7.

Links

• Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 1..104 (first 65 terms from Alois P. Heinz)

Crossrefs

Column k=3 of A196231.

Cf. A161943, A164949, A232534.

Programs

Maple

b:= proc(n, k, i) option remember; local m;
      m:= i*(i+1)/2;
      if k>n then b(k, n, i)
    elif k>=0 and n+k>m or k<0 and n-2*k>m then 0
    elif [n, k, i] = [0, 0, 0] then 1
    else b(n, k, i-1)+b(n+i, k+i, i-1)+b(n-i, k, i-1)+b(n, k-i, i-1)
      fi
    end:
a:= proc(n) option remember;
      `if`(n>2, b(n, n, n-1)/2+ a(n-1), 0)
    end:
seq(a(n), n=1..20);

Mathematica

b[n_, k_, i_] := b[n, k, i] = Module[{m = i*(i+1)/2}, Which[k>n , b[k, n, i], k >= 0 && n+k>m || k<0 && n-2*k > m, 0, {n, k, i} == {0, 0, 0}, 1, True, b[n, k, i-1] + b[n+i, k+i, i-1] + b[n-i, k, i-1] + b[n, k-i, i-1]]]; a[n_] := a[n] = If[n>2, b[n, n, n-1]/2 + a[n-1], 0]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Feb 05 2015, after Alois P. Heinz *)

Formula

Conjecture: a(n) ~ 4^n / (Pi * sqrt(3) * n^3). - Vaclav Kotesovec, Oct 16 2014