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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.
2
1, 1, 3, 6, 19, 53, 225, 878, 4281, 21212, 117489, 678571, 4238024, 27644438, 191326221, 1383029112, 10490101937, 82864869805, 682358388107, 5832742205058, 51733248275075, 474870253871245, 4507061060486642, 44152005855084347, 445973953222607799
OFFSET
0,3
COMMENTS
All block lengths and n are multiples of the number of available colors d.
LINKS
FORMULA
a(n) = Sum_{d|n} A275043(n/d,d) for n > 0, a(0) = 1.
a(p) = 1 + A000110(p) for prime p.
EXAMPLE
a(0) = 1: (), the empty partition.
a(1) = 1: 1a.
a(2) = 3: 1a2b, 1a2a, 1a|2a.
a(3) = 6: 1a2b3c, 1a2a3a, 1a2a|3a, 1a3a|2a, 1a|2a3a, 1a|2a|3a.
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.
Here the colors a, b, c, ... are used.
MAPLE
A:= proc(n, k) option remember; `if`(n=0, 1, add(
binomial(n, j)^k*(n-j)*A(j, k), j=0..n-1)/n)
end:
a:= n-> `if`(n=0, 1, add(A(n/d, d), d=numtheory[divisors](n))):
seq(a(n), n=0..28);
MATHEMATICA
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];
a[n_] := If[n == 0, 1, Sum[A[n/d, d], {d, Divisors[n]}]];
Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Aug 25 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 17 2021
STATUS
approved