A370364 - OEIS

%I #13 Feb 17 2024 15:25:47

%S 0,1,1,28,22893,2443061876,68542265471953355,

%T 833412961429901104030214430,6514551431426932053792271970458170132097,

%U 45458343253887079540702419310885199704811913950207054152,375236832464739513549091449370258959406125572044428827214970469920572831639

%N Number of partitions of [n^2] into n sets of size n having at least one set of consecutive numbers whose maximum (if n>0) is a multiple of n.

%H Alois P. Heinz, <a href="/A370364/b370364.txt">Table of n, a(n) for n = 0..27</a>

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

%F a(n) = Sum_{j=0..n-1} (-1)^(n-j+1)*binomial(n,j)*(n*j)!/(j!*n!^j).

%F a(n) = A370363(n,n).

%F a(n) = A057599(n) - A370367(n).

%e a(1) = 1: 1.

%e a(2) = 1: 12|34.

%e a(3) = 28: 123|456|789, 123|457|689, 123|458|679, 123|459|678, 123|467|589, 123|468|579, 123|469|578, 123|478|569, 123|479|568, 123|489|567, 124|356|789, 125|346|789, 126|345|789, 127|389|456, 128|379|456, 129|378|456, 134|256|789, 135|246|789, 136|245|789, 137|289|456, 138|279|456, 139|278|456, 145|236|789, 146|235|789, 156|234|789, 178|239|456, 179|238|456, 189|237|456.

%p a:= n-> add((-1)^(n-j+1)*binomial(n, j)*(n*j)!/(j!*n!^j), j=0..n-1):

%p seq(a(n), n=0..10);

%Y Main diagonal of A370363.

%Y Cf. A057599, A370367.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Feb 16 2024