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A275422
Number A(n,k) of set partitions of [n] such that k is a multiple of each block size; square array A(n,k), n>=0, k>=0, read by antidiagonals.
10
1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 1, 15, 1, 1, 1, 4, 1, 52, 1, 1, 2, 2, 10, 1, 203, 1, 1, 1, 4, 5, 26, 1, 877, 1, 1, 2, 1, 11, 11, 76, 1, 4140, 1, 1, 1, 5, 1, 31, 31, 232, 1, 21147, 1, 1, 2, 1, 14, 2, 106, 106, 764, 1, 115975, 1, 1, 1, 4, 1, 46, 7, 372, 337, 2620, 1, 678570
OFFSET
0,6
LINKS
FORMULA
E.g.f. for column k>0: exp(Sum_{d|k} x^d/d!), for k=0: exp(exp(x)-1).
EXAMPLE
A(5,3) = 11: 123|4|5, 124|3|5, 125|3|4, 134|2|5, 135|2|4, 1|234|5, 1|235|4, 145|2|3, 1|245|3, 1|2|345, 1|2|3|4|5.
A(4,4) = 11: 1234, 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
A(6,5) = 7: 12345|6, 12346|5, 12356|4, 12456|3, 13456|2, 1|23456, 1|2|3|4|5|6.
Square array A(n,k) begins:
: 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
: 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
: 2, 1, 2, 1, 2, 1, 2, 1, 2, ...
: 5, 1, 4, 2, 4, 1, 5, 1, 4, ...
: 15, 1, 10, 5, 11, 1, 14, 1, 11, ...
: 52, 1, 26, 11, 31, 2, 46, 1, 31, ...
: 203, 1, 76, 31, 106, 7, 167, 1, 106, ...
: 877, 1, 232, 106, 372, 22, 659, 2, 372, ...
: 4140, 1, 764, 337, 1499, 57, 2836, 9, 1500, ...
MAPLE
A:= proc(n, k) option remember; `if`(n=0, 1, add(
`if`(j>n, 0, A(n-j, k)*binomial(n-1, j-1)), j=
`if`(k=0, 1..n, numtheory[divisors](k))))
end:
seq(seq(A(n, d-n), n=0..d), d=0..14);
MATHEMATICA
A[n_, k_] := A[n, k] = If[n==0, 1, Sum[If[j>n, 0, A[n-j, k]*Binomial[n-1, j - 1]], {j, If[k==0, Range[n], Divisors[k]]}]]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 08 2017, translated from Maple *)
CROSSREFS
Main diagonal gives A275429.
Sequence in context: A213945 A290771 A014651 * A169951 A174453 A361781
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 27 2016
STATUS
approved