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A245405
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Number A(n,k) of endofunctions on [n] such that no element has a preimage of cardinality k; 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, 0, 2, 1, 1, 2, 6, 1, 1, 2, 3, 24, 1, 1, 4, 9, 40, 120, 1, 1, 4, 24, 76, 205, 720, 1, 1, 4, 27, 208, 825, 2556, 5040, 1, 1, 4, 27, 252, 2325, 10206, 24409, 40320, 1, 1, 4, 27, 256, 3025, 31956, 143521, 347712, 362880, 1, 1, 4, 27, 256, 3120, 44406, 520723, 2313200, 4794633, 3628800
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OFFSET
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0,6
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LINKS
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FORMULA
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A(n,k) = n! * [x^n] (exp(x)-x^k/k!)^n.
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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, 0, 1, 1, 1, 1, 1, ...
2 : 2, 2, 2, 4, 4, 4, 4, ...
3 : 6, 3, 9, 24, 27, 27, 27, ...
4 : 24, 40, 76, 208, 252, 256, 256, ...
5 : 120, 205, 825, 2325, 3025, 3120, 3125, ...
6 : 720, 2556, 10206, 31956, 44406, 46476, 46650, ...
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MAPLE
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b:= proc(n, i, k) option remember; `if`(n=0 and i=0, 1,
`if`(i<1, 0, add(`if`(j=k, 0, b(n-j, i-1, k)*
binomial(n, j)), j=0..n)))
end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);
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MATHEMATICA
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nn = n; f[m_]:=Flatten[Table[m[[j, i - j + 1]], {i, 1, Length[m]}, {j, 1, i}]]; f[Transpose[Table[Prepend[Table[n! Coefficient[Series[(Exp[x] -x^k/k!)^n, {x, 0, nn}], x^n], {n, 1, 10}], 1], {k, 0, 10}]]] (* Geoffrey Critzer, Jan 31 2015 *)
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CROSSREFS
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Column k=0-10 give: A000142, A231797, A245406, A245407, A245408, A245409, A245410, A245411, A245412, A245413, A245414.
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KEYWORD
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AUTHOR
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STATUS
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approved
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