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Number A(n,k) of set partitions of [n] such that i-j is a multiple of k for all i,j belonging to the same block; square array A(n,k), n>=0, k>=0, read by antidiagonals.
10

%I #14 Oct 18 2018 16:55:46

%S 1,1,1,1,1,1,1,1,2,1,1,1,1,5,1,1,1,1,2,15,1,1,1,1,1,4,52,1,1,1,1,1,2,

%T 10,203,1,1,1,1,1,1,4,25,877,1,1,1,1,1,1,2,8,75,4140,1,1,1,1,1,1,1,4,

%U 20,225,21147,1,1,1,1,1,1,1,2,8,50,780,115975,1

%N Number A(n,k) of set partitions of [n] such that i-j is a multiple of k for all i,j belonging to the same block; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%H Alois P. Heinz, <a href="/A275069/b275069.txt">Antidiagonals n = 0..140, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%F A(n,k) = Product_{i=0..k-1} A000110(floor((n+i)/k)).

%e A(5,0) = 1: 1|2|3|4|5.

%e A(5,1) = 52 = A000110(5).

%e A(5,2) = 10: 135|24, 13|24|5, 135|2|4, 13|2|4|5, 15|24|3, 1|24|35, 1|24|3|5, 15|2|3|4, 1|2|35|4, 1|2|3|4|5.

%e A(5,3) = 4: 14|25|3, 14|2|3|5, 1|25|3|4, 1|2|3|4|5.

%e A(5,4) = 2: 15|2|3|4, 1|2|3|4|5.

%e Square array A(n,k) begins:

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 5, 2, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 15, 4, 2, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 52, 10, 4, 2, 1, 1, 1, 1, 1, 1, ...

%e 1, 203, 25, 8, 4, 2, 1, 1, 1, 1, 1, ...

%e 1, 877, 75, 20, 8, 4, 2, 1, 1, 1, 1, ...

%e 1, 4140, 225, 50, 16, 8, 4, 2, 1, 1, 1, ...

%e 1, 21147, 780, 125, 40, 16, 8, 4, 2, 1, 1, ...

%e 1, 115975, 2704, 375, 100, 32, 16, 8, 4, 2, 1, ...

%p with(combinat):

%p A:= (n, k)-> mul(bell(floor((n+i)/k)), i=0..k-1):

%p seq(seq(A(n, d-n), n=0..d), d=0..14);

%t A[n_, k_] := Product[BellB[Floor[(n+i)/k]], {i, 0, k-1}]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Feb 17 2017, translated from Maple *)

%Y Columns k=0-10 give: A000012, A000110, A124419, A275070, A275071, A275072, A275073, A275074, A275075, A275076, A275077.

%Y A(k*n,n) for k=1-4 gives: A000012, A000079, A000351, A001024.

%K nonn,tabl

%O 0,9

%A _Alois P. Heinz_, Jul 15 2016