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A370363 Number A(n,k) of partitions of [k*n] into n sets of size k having at least one set of consecutive numbers whose maximum (if k>0) is a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A370363 #21 Feb 19 2024 18:28:37
%S A370363 0,0,1,0,1,1,0,1,1,1,0,1,1,1,1,0,1,1,7,1,1,0,1,1,28,45,1,1,0,1,1,103,
%T A370363 1063,401,1,1,0,1,1,376,22893,74296,4355,1,1,0,1,1,1384,503751,
%U A370363 13080721,8182855,56127,1,1,0,1,1,5146,11432655,2443061876,15237712355,1305232804,836353,1,1
%N A370363 Number A(n,k) of partitions of [k*n] into n sets of size k having at least one set of consecutive numbers whose maximum (if k>0) is a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A370363 Alois P. Heinz, <a href="/A370363/b370363.txt">Antidiagonals n = 0..55, flattened</a>
%H A370363 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A370363 A(n,k) = A060540(n,k) - A370366(n,k) for n,k >= 1.
%e A370363 A(3,2) = 7: 12|34|56, 12|35|46, 12|36|45, 13|24|56, 14|23|56, 15|26|34, 16|25|34.
%e A370363 Square array A(n,k) begins:
%e A370363   0, 0,   0,     0,        0,          0, ...
%e A370363   1, 1,   1,     1,        1,          1, ...
%e A370363   1, 1,   1,     1,        1,          1, ...
%e A370363   1, 1,   7,    28,      103,        376, ...
%e A370363   1, 1,  45,  1063,    22893,     503751, ...
%e A370363   1, 1, 401, 74296, 13080721, 2443061876, ...
%p A370363 A:= proc(n, k) option remember; `if`(k=0, signum(n), add(
%p A370363       (-1)^(n-j+1)*binomial(n, j)*(k*j)!/(j!*k!^j), j=0..n-1))
%p A370363     end:
%p A370363 seq(seq(A(n, d-n), n=0..d), d=0..10);
%Y A370363 Columns k=0+1,2-3 give: A057427, A370253, A370358.
%Y A370363 Rows n=0,1+2,3 give: A000004, A000012, A370487.
%Y A370363 Main diagonal gives A370364.
%Y A370363 Antidiagonal sums give A370365.
%Y A370363 Cf. A060540, A370366.
%K A370363 nonn,tabl
%O A370363 0,19
%A A370363 _Alois P. Heinz_, Feb 16 2024