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A206823
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Triangular array read by rows: T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} with exactly k elements x such that |f^(-1)(x)| = 1; n>=0, 0<=k<=n.
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3
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1, 0, 1, 2, 0, 2, 3, 18, 0, 6, 40, 48, 144, 0, 24, 205, 1000, 600, 1200, 0, 120, 2556, 7380, 18000, 7200, 10800, 0, 720, 24409, 125244, 180810, 294000, 88200, 105840, 0, 5040, 347712, 1562176, 4007808, 3857280, 4704000, 1128960, 1128960, 0, 40320
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OFFSET
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0,4
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COMMENTS
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Row sums = n^n, all functions f:{1,2,...,n}->{1,2,...,n}.
T(n,n)= n!, bijections on {1,2,...,n}.
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LINKS
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FORMULA
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E.g.f.: Sum_{k=0..n} T(n,k) * y^k * x^n / n! = (exp(x) - x + y*x)^n.
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EXAMPLE
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Triangle T(n,k) begins:
1;
0 1;
2 0 2;
3 18 0 6;
40 48 144 0 24;
205 1000 600 1200 0 120;
...
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MAPLE
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with(combinat): C:= binomial:
b:= proc(t, i, u) option remember; `if`(t=0, 1,
`if`(i<2, 0, b(t, i-1, u) +add(multinomial(t, t-i*j, i$j)
*b(t-i*j, i-1, u-j)*u!/(u-j)!/j!, j=1..t/i)))
end:
T:= (n, k)-> C(n, k)*C(n, k)*k! *b(n-k$2, n-k):
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MATHEMATICA
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nn = 8; Prepend[CoefficientList[Table[n! Coefficient[Series[(Exp[x] - x + y x)^n, {x, 0, nn}], x^n], {n, 1, nn}], y], {1}] // Flatten
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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