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A345400 Number of colored set partitions of [n] where (i mod d) identifies the color of i and d is the number of available colors such that within each block the frequency of all colors is equal.

%I A345400 #18 Aug 25 2021 09:02:48
%S A345400 1,1,3,6,19,53,225,878,4281,21212,117489,678571,4238024,27644438,
%T A345400 191326221,1383029112,10490101937,82864869805,682358388107,
%U A345400 5832742205058,51733248275075,474870253871245,4507061060486642,44152005855084347,445973953222607799
%N A345400 Number of colored set partitions of [n] where (i mod d) identifies the color of i and d is the number of available colors such that within each block the frequency of all colors is equal.
%C A345400 All block lengths and n are multiples of the number of available colors d.
%H A345400 Alois P. Heinz, <a href="/A345400/b345400.txt">Table of n, a(n) for n = 0..576</a>
%H A345400 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A345400 a(n) = Sum_{d|n} A275043(n/d,d) for n > 0, a(0) = 1.
%F A345400 a(p) = 1 + A000110(p) for prime p.
%e A345400 a(0) = 1: (), the empty partition.
%e A345400 a(1) = 1: 1a.
%e A345400 a(2) = 3: 1a2b, 1a2a, 1a|2a.
%e A345400 a(3) = 6: 1a2b3c, 1a2a3a, 1a2a|3a, 1a3a|2a, 1a|2a3a, 1a|2a|3a.
%e A345400 a(4) = 19: 1a2b3c4d, 1a2b3a4b, 1a2b|3a4b, 1a4b|2b3a, 1a2a3a4a, 1a2a3a|4a, 1a2a4a|3a, 1a2a|3a4a, 1a2a|3a|4a, 1a3a4a|2a, 1a3a|2a4a, 1a3a|2a|4a, 1a4a|2a3a, 1a|2a3a4a, 1a|2a3a|4a, 1a4a|2a|3a, 1a|2a4a|3a, 1a|2a|3a4a, 1a|2a|3a|4a.
%e A345400 Here the colors a, b, c, ... are used.
%p A345400 A:= proc(n, k) option remember; `if`(n=0, 1, add(
%p A345400       binomial(n, j)^k*(n-j)*A(j, k), j=0..n-1)/n)
%p A345400     end:
%p A345400 a:= n-> `if`(n=0, 1, add(A(n/d, d), d=numtheory[divisors](n))):
%p A345400 seq(a(n), n=0..28);
%t A345400 A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Binomial[n, j]^k*(n - j)*A[j, k], {j, 0, n - 1}]/n];
%t A345400 a[n_] := If[n == 0, 1, Sum[A[n/d, d], {d, Divisors[n]}]];
%t A345400 Table[a[n], {n, 0, 28}] (* _Jean-François Alcover_, Aug 25 2021, after _Alois P. Heinz_ *)
%Y A345400 Cf. A000040, A000110, A275043.
%K A345400 nonn
%O A345400 0,3
%A A345400 _Alois P. Heinz_, Jun 17 2021