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%I A275044 #14 Jul 12 2020 10:47:50
%S A275044 1,1,3,64,25097,350813126,293327384637282,22208366234650578141209,
%T A275044 213426677887357366350726096998529,
%U A275044 344735749788852590196707169431958672823413322,118966637603805785518622376062965559343297730169187276656138
%N A275044 Number of set partitions of [n^2] such that within each block the numbers of elements from all residue classes modulo n are equal for n>0, a(0)=1.
%H A275044 Alois P. Heinz, <a href="/A275044/b275044.txt">Table of n, a(n) for n = 0..30</a>
%H A275044 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A275044 a(n) = (n!)^n * [x^n] exp(Sum_{k>=1} x^k / (k!)^n). - _Ilya Gutkovskiy_, Jul 12 2020
%e A275044 a(2) = 3: 1234, 12|34, 14|23.
%e A275044 a(3) = 64: 123456789, 123456|789, 123459|678, 123468|579, ... , 159|267|348, 168|279|345, 189|267|345.
%p A275044 b:= proc(n, k) option remember; `if`(k*n=0, 1, add(
%p A275044        binomial(n, j)^k*(n-j)*b(j, k), j=0..n-1)/n)
%p A275044     end:
%p A275044 a:= n-> b(n$2):
%p A275044 seq(a(n), n=0..12);
%t A275044 b[n_, k_] := b[n, k] = If[k*n == 0, 1, Sum[Binomial[n, j]^k*(n-j)*b[j, k], {j, 0, n-1}]/n];
%t A275044 a[n_] := b[n, n];
%t A275044 Table[a[n], {n, 0, 12}] (* _Jean-François Alcover_, May 27 2018, translated from Maple *)
%Y A275044 Main diagonal of A275043.
%K A275044 nonn
%O A275044 0,3
%A A275044 _Alois P. Heinz_, Jul 14 2016

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