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A245910
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Number A(n,k) of pairs of endofunctions f, g on [n] satisfying f(g^k(i)) = f(i) for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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14
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1, 1, 1, 1, 1, 16, 1, 1, 10, 729, 1, 1, 12, 159, 65536, 1, 1, 10, 249, 3496, 9765625, 1, 1, 12, 207, 7744, 98345, 2176782336, 1, 1, 10, 249, 6856, 326745, 3373056, 678223072849, 1, 1, 12, 159, 9184, 302345, 17773056, 136535455, 281474976710656
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OFFSET
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0,6
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LINKS
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EXAMPLE
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Square array A(n,k) begins:
0 : 1, 1, 1, 1, 1, 1, ...
1 : 1, 1, 1, 1, 1, 1, ...
2 : 16, 10, 12, 10, 12, 10, ...
3 : 729, 159, 249, 207, 249, 159, ...
4 : 65536, 3496, 7744, 6856, 9184, 3496, ...
5 : 9765625, 98345, 326745, 302345, 488745, 173225, ...
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MAPLE
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with(combinat):
b:= proc(n, i, k) option remember; unapply(`if`(n=0 or i=1, x^n,
expand(add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
x^(igcd(i, k)*j)*b(n-i*j, i-1, k)(x), j=0..n/i))), x)
end:
A:= (n, k)-> `if`(k=0, n^(2*n), add(binomial(n-1, j-1)*n^(n-j)*
b(j$2, k)(n), j=0..n)):
seq(seq(A(n, d-n), n=0..d), d=0..10);
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MATHEMATICA
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multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, k_] := b[n, i, k] = Function[{x}, If[n == 0 || i == 1, x^n, Expand[Sum[(i-1)!^j*multinomial[n, Join[{ n-i*j}, Array[i&, j]]]/j!*x^(GCD[i, k]*j)*b[n-i*j, i-1, k][x], {j, 0, n/i}]]]]; A[0, _] = 1; A[n_, k_] := If[k == 0, n^(2n), Sum[Binomial[n-1, j-1]*n^(n-j)* b[j, j, k][n], {j, 0, n}]]; Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 04 2015, after Alois P. Heinz *)
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CROSSREFS
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Columns k=0-10 give: A062206, A239761, A239777, A245912, A245913, A245914, A245915, A245916, A245917, A245918, A245919.
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KEYWORD
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AUTHOR
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STATUS
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approved
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