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A246189
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Number of endofunctions on [n] where the smallest cycle length equals 2.
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2
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1, 6, 51, 580, 8265, 141246, 2810437, 63748728, 1622579985, 45775778950, 1417347491241, 47776074289164, 1741386177576409, 68238497945688630, 2860625245955274225, 127736893134458097136, 6052712065187733972513, 303322427195785592735502, 16028016368907840953165425
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OFFSET
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2,2
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LINKS
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FORMULA
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MAPLE
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with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>n, 0,
add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
b(n-i*j, i+1), j=0..n/i)))
end:
A:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j, k), j=0..n):
a:= n-> A(n, 2) -A(n, 3):
seq(a(n), n=2..25);
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MATHEMATICA
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multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = If[n==0, 1, If[i>n, 0, Sum[(i - 1)!^j multinomial[ n, Join[{n - i*j}, Table[i, {j}]]]/j! b[n - i*j, i + 1], {j, 0, n/i}]]];
A[n_, k_] := Sum[Binomial[n - 1, j - 1] n^(n - j) b[j, k], {j, 0, n}];
a[n_] := A[n, 2] - A[n, 3];
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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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