A394818 Number of set partitions of [n] such that the elements of the k-th block can each be k different colors.

1, 1, 3, 15, 111, 1119, 14583, 236775, 4659951, 108835599, 2965927143, 93015145335, 3318634499871, 133400225472159, 5991382911276183, 298493219945303175, 16392036941979061071, 986729309825162666799, 64784791274694379877703, 4618773443181937984415895

Offset

0,3

Comments

Here the blocks of a given set partition are ordered by their least element.

All terms are odd.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..319

Formula

G.f.: A(y) = Sum_{i>=0} i!*y^i / Product_{j=1..i} (1 - y*j*(j+1)/2).

Example

a(3) = 15: {{1,2,3}}, {{1,2},{3_a}}, {{1,2},{3_b}}, {{1,3},{2_a}}, {{1,3},{2_b}}, {{1},
{2_a,2_a}}, {{1},{2_a,2_b}}, {{1},{2_b,2_a}}, {{1},{2_b,2_b}}, {{1},{2_a},{3_a}}, {{1},{2_a},{3_b}}, {{1},{2_a},{3_c}}, {{1},{2_b},{3_a}}, {{1},{2_b},{3_b}}, {{1},{2_b},{3_c}}.

Maple

b:= proc(n, m) option remember; `if`(n=0, 1,
      (b(n-1, m)*m/2+b(n-1, m+1))*(m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..19);  # Alois P. Heinz, May 02 2026

Prog

(PARI) A_y(N) = {my(y='y+O('y^(N+1)), A = sum(i=0, N,  i!*y^i / prod(j=1, i, 1 - y*j*(j+1)/2))); Vec(A)}

Crossrefs

Column sums of A394777.

Cf. A000110, A088305, A383253.

Sequence in context: A370877 A375905 A254789 * A112936 A001063 A130168

Adjacent sequences: A394815 A394816 A394817 * A394819 A394820 A394821

Keyword

nonn,easy

Author

John Tyler Rascoe, May 02 2026

Status

approved