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A275044
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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.
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5
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1, 1, 3, 64, 25097, 350813126, 293327384637282, 22208366234650578141209, 213426677887357366350726096998529, 344735749788852590196707169431958672823413322, 118966637603805785518622376062965559343297730169187276656138
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = (n!)^n * [x^n] exp(Sum_{k>=1} x^k / (k!)^n). - Ilya Gutkovskiy, Jul 12 2020
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EXAMPLE
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a(2) = 3: 1234, 12|34, 14|23.
a(3) = 64: 123456789, 123456|789, 123459|678, 123468|579, ... , 159|267|348, 168|279|345, 189|267|345.
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MAPLE
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b:= proc(n, k) option remember; `if`(k*n=0, 1, add(
binomial(n, j)^k*(n-j)*b(j, k), j=0..n-1)/n)
end:
a:= n-> b(n$2):
seq(a(n), n=0..12);
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MATHEMATICA
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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];
a[n_] := b[n, n];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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