A394818 - OEIS

%I #17 May 12 2026 21:06:28

%S 1,1,3,15,111,1119,14583,236775,4659951,108835599,2965927143,

%T 93015145335,3318634499871,133400225472159,5991382911276183,

%U 298493219945303175,16392036941979061071,986729309825162666799,64784791274694379877703,4618773443181937984415895

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

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

%C All terms are odd.

%H Alois P. Heinz, <a href="/A394818/b394818.txt">Table of n, a(n) for n = 0..319</a>

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

%e a(3) = 15: {{1,2,3}}, {{1,2},{3_a}}, {{1,2},{3_b}}, {{1,3},{2_a}}, {{1,3},{2_b}}, {{1},

%e {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}}.

%p b:= proc(n, m) option remember; `if`(n=0, 1,

%p       (b(n-1, m)*m/2+b(n-1, m+1))*(m+1))

%p     end:

%p a:= n-> b(n, 0):

%p seq(a(n), n=0..19);  # _Alois P. Heinz_, May 02 2026

%o (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)}

%Y Column sums of A394777.

%Y Cf. A000110, A088305, A383253.

%K nonn,easy

%O 0,3

%A _John Tyler Rascoe_, May 02 2026