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A275069 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.

%I A275069 #14 Oct 18 2018 16:55:46
%S A275069 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 A275069 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 A275069 20,225,21147,1,1,1,1,1,1,1,2,8,50,780,115975,1
%N A275069 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 A275069 Alois P. Heinz, <a href="/A275069/b275069.txt">Antidiagonals n = 0..140, flattened</a>
%H A275069 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A275069 A(n,k) = Product_{i=0..k-1} A000110(floor((n+i)/k)).
%e A275069 A(5,0) = 1: 1|2|3|4|5.
%e A275069 A(5,1) = 52 = A000110(5).
%e A275069 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 A275069 A(5,3) = 4: 14|25|3, 14|2|3|5, 1|25|3|4, 1|2|3|4|5.
%e A275069 A(5,4) = 2: 15|2|3|4, 1|2|3|4|5.
%e A275069 Square array A(n,k) begins:
%e A275069   1,      1,    1,   1,   1,  1,  1, 1, 1, 1, 1, ...
%e A275069   1,      1,    1,   1,   1,  1,  1, 1, 1, 1, 1, ...
%e A275069   1,      2,    1,   1,   1,  1,  1, 1, 1, 1, 1, ...
%e A275069   1,      5,    2,   1,   1,  1,  1, 1, 1, 1, 1, ...
%e A275069   1,     15,    4,   2,   1,  1,  1, 1, 1, 1, 1, ...
%e A275069   1,     52,   10,   4,   2,  1,  1, 1, 1, 1, 1, ...
%e A275069   1,    203,   25,   8,   4,  2,  1, 1, 1, 1, 1, ...
%e A275069   1,    877,   75,  20,   8,  4,  2, 1, 1, 1, 1, ...
%e A275069   1,   4140,  225,  50,  16,  8,  4, 2, 1, 1, 1, ...
%e A275069   1,  21147,  780, 125,  40, 16,  8, 4, 2, 1, 1, ...
%e A275069   1, 115975, 2704, 375, 100, 32, 16, 8, 4, 2, 1, ...
%p A275069 with(combinat):
%p A275069 A:= (n, k)-> mul(bell(floor((n+i)/k)), i=0..k-1):
%p A275069 seq(seq(A(n, d-n), n=0..d), d=0..14);
%t A275069 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 A275069 Columns k=0-10 give: A000012, A000110, A124419, A275070, A275071, A275072, A275073, A275074, A275075, A275076, A275077.
%Y A275069 A(k*n,n) for k=1-4 gives: A000012, A000079, A000351, A001024.
%K A275069 nonn,tabl
%O A275069 0,9
%A A275069 _Alois P. Heinz_, Jul 15 2016