A365441 Number of partitions of [2n] whose block maxima sum to 3n.

1, 1, 2, 5, 10, 23, 47, 103, 209, 449, 908, 1909, 3864, 8011, 16165, 33244, 66933, 136628, 274876, 558107, 1121160, 2268526, 4552291, 9183569, 18417449, 37073504, 74300048, 149334422, 299134695, 600481001, 1202436958, 2411536369, 4827532935, 9675363921, 19364235775

Offset: 0

Author

Alois P. Heinz, Dec 06 2023

Keywords

• nonn

Comments

Rows of A367955 and the reversed columns of A367955 converge to this sequence.

Examples

			a(0) = 1: the empty partition.
a(1) = 1: 1|2.
a(2) = 2: 12|34, 134|2.
a(3) = 5: 123|456, 12456|3, 13|2456, 1456|23, 1|2|3456.
a(4) = 10: 1234|5678, 1235678|4, 124|35678, 125678|34, 134|25678, 135678|24, 14|235678, 15678|234, 1|23|45678, 1|245678|3.
a(5) = 23: 12345|6789(10), 12346789(10)|5, 1235|46789(10), 1236789(10)|45, 1245|36789(10), 1246789(10)|35, 125|346789(10), 126789(10)|345, 12|3|456789(10), 1345|26789(10), 1346789(10)|25, 135|246789(10), 136789(10)|245, 13|2|456789(10), 145|236789(10), 146789(10)|235, 15|2346789(10), 16789(10)|2345, 1|234|56789(10), 1|2356789(10)|4, 1456789(10)|2|3, 1|24|356789(10), 1|256789(10)|34.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3321
• Wikipedia, Partition of a set

Crossrefs

Cf. A000110, A367955, A367969, A368204, A368675.

Programs

Maple

b:= proc(n, m) option remember; `if`(n=0, 1,
      b(n-1, m)*m + expand(x^n*b(n-1, m+1)))
    end:
a:= n-> coeff(b(2*n, 0), x, 3*n):
seq(a(n), n=0..42);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(i*(i+1)/2 b(3*n, 2*n, 0):
seq(a(n), n=0..42);

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[i*(i + 1)/2 < n, 0, If[n == 0, t^i, If[t == 0, 0, t*b[n, i - 1, t]] + (t + 1)^Max[0, 2*i - n - 1]*b[n - i, Min[n - i, i - 1], t + 1]]];
a[n_] := If[n == 0, 1, b[3n, 2n, 0]];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Oct 03 2024, after Alois P. Heinz *)

Formula

a(n) = A367955(2n,3n).