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

1,6

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.

Links

Table of n, a(n) for n=1..25.

Example

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}.

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 *)

Crossrefs

Cf. A164934, A232466 (2 blocks).

Column k=3 of A248112.

Sequence in context: A162036 A321253 A290073 * A274594 A062422 A320553

Adjacent sequences: A232531 A232532 A232533 * A232535 A232536 A232537

Keyword

nonn,more

Author

Alois P. Heinz, Nov 25 2013

Extensions

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

Status

approved