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A374319 Number of partitions of [n] such that the number of blocks of size k is zero or a divisor of k for every k.

%I A374319 #18 Jul 04 2024 20:16:07
%S A374319 1,1,1,4,8,31,82,274,1626,5135,26751,125489,1020692,4333707,31083613,
%T A374319 132960104,1323145731,8282668312,70017330978,423293287673,
%U A374319 3135764479898,30762429056580,269133472001923,2185746568531948,15121514389566421,147045774699171957
%N A374319 Number of partitions of [n] such that the number of blocks of size k is zero or a divisor of k for every k.
%H A374319 Alois P. Heinz, <a href="/A374319/b374319.txt">Table of n, a(n) for n = 0..598</a>
%H A374319 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A374319 a(0) = 1: the empty partition.
%e A374319 a(1) = 1: 1.
%e A374319 a(2) = 1: 12.
%e A374319 a(3) = 4: 123, 12|3, 13|2, 1|23.
%e A374319 a(4) = 8: 1234, 123|4, 124|3, 12|34, 134|2, 13|24, 14|23, 1|234.
%e A374319 a(5) = 31: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|34|5, 12|35|4, 12|3|45, 1345|2, 134|25, 135|24, 13|245, 13|24|5, 13|25|4, 13|2|45, 145|23, 14|235, 14|23|5, 15|234, 1|2345, 15|23|4, 1|23|45, 14|25|3, 14|2|35, 15|24|3, 1|24|35, 15|2|34, 1|25|34.
%p A374319 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A374319       add(`if`(j=0 or irem(i, j)=0, b(n-i*j, i-1)/j!*
%p A374319       combinat[multinomial](n, i$j, n-i*j), 0), j=0..n/i)))
%p A374319     end:
%p A374319 a:= n-> b(n$2):
%p A374319 seq(a(n), n=0..27);
%Y A374319 Cf. A374292, A374320, A374321, A374329.
%K A374319 nonn
%O A374319 0,4
%A A374319 _Alois P. Heinz_, Jul 04 2024