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A246106
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Number A(n,k) of inequivalent n X n matrices with entries from [k], where equivalence means permutations of rows or columns; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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30
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1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 7, 1, 0, 1, 4, 27, 36, 1, 0, 1, 5, 76, 738, 317, 1, 0, 1, 6, 175, 8240, 90492, 5624, 1, 0, 1, 7, 351, 57675, 7880456, 64796982, 251610, 1, 0, 1, 8, 637, 289716, 270656150, 79846389608, 302752867740, 33642660, 1, 0
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OFFSET
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0,8
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LINKS
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FORMULA
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A(n,k) = Sum_{i=0..k} C(k,i) * A256069(n,i).
A(n,k) = Sum_{p,q in P(n)} k^Sum_{i in p, j in q} gcd(i, j) / (N(p)*N(q)) where N(p) = Sum_{distinct parts x in p} x^m(x)*m(x)!, m(x) = multiplicity of x in p. - M. F. Hasler, Apr 30 2022
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EXAMPLE
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Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 1, 7, 27, 76, 175, ...
0, 1, 36, 738, 8240, 57675, ...
0, 1, 317, 90492, 7880456, 270656150, ...
0, 1, 5624, 64796982, 79846389608, 20834113243925, ...
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, [[]],
`if`(i<1, [], [b(n, i-1)[], seq(map(p->[p[], [i, j]],
b(n-i*j, i-1))[], j=1..n/i)]))
end:
A:= proc(n, k) option remember; add(add(k^add(add(i[2]*j[2]*
igcd(i[1], j[1]), j=t), i=s) /mul(i[1]^i[2]*i[2]!, i=s)
/mul(i[1]^i[2]*i[2]!, i=t), t=b(n$2)), s=b(n$2))
end:
seq(seq(A(n, d-n), n=0..d), d=0..10);
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PROG
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CROSSREFS
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Columns k = 0-10 give: A000007, A000012, A002724, A052269, A052271, A052272, A246112, A246113, A246114, A246115, A246116.
Rows n = 0-10 give: A000012, A001477, A039623, A058001, A058002, A058003, A058004, A246108, A246109, A246110, A246111.
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KEYWORD
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AUTHOR
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STATUS
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approved
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