A394281 Number of partitions of [n] whose set of block sizes equals [k] for k = A003056(n).

1, 1, 1, 3, 6, 25, 60, 210, 1400, 7560, 12600, 69300, 693000, 5405400, 46666620, 37837800, 302702400, 4288284000, 46313467200, 542639457360, 5768599636800, 2053230379200, 22585534171200, 432889404948000, 6233607431251200, 96101447898456000, 1328104694380464000

Offset

0,4

Comments

k = A003056(n) is the largest integer such that the k-th triangular number A000217(k) is <= n.

The sequence is not monotonic: a(n) < a(n-1) for all triangular numbers n >= 15.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..685

Wikipedia, Partition of a set

Formula

a(A000217(n)) = A022915(n).

a(n) <= A393098(n).

Example

a(4) = 6: 12|3|4, 13|2|4, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
a(5) = 25: 12|34|5, 12|35|4, 12|3|45, 12|3|4|5, 13|24|5, 13|25|4, 13|2|45, 13|2|4|5, 14|23|5, 15|23|4, 1|23|45, 1|23|4|5, 14|25|3, 14|2|35, 14|2|3|5, 15|24|3, 1|24|35, 1|24|3|5, 15|2|34, 1|25|34, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45.

Maple

f:= proc(n) option remember; floor((sqrt(1+8*n)-1)/2) end:
b:= proc(n, i) option remember; uses combinat; `if`(n=0,
      `if`(i=0, 1, 0), add(b(n-i*j, i-1)*multinomial
          (n, n-i*j, i$j)/j!, j=`if`(i=1, n, 1..n/i)))
    end:
a:= n-> b(n, f(n)):
seq(a(n), n=0..26);

Crossrefs

Cf. A000110, A000217, A003056, A022915, A393098.

Last elements of rows in A387082.

Sequence in context: A288760 A144227 A174473 * A148657 A148658 A148659

Adjacent sequences: A394278 A394279 A394280 * A394282 A394283 A394284

Keyword

nonn

Author

Alois P. Heinz, Apr 13 2026

Status

approved