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A055134
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Triangle read by rows: T(n,k) = number of labeled endofunctions on n points with k fixed points.
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9
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1, 0, 1, 1, 2, 1, 8, 12, 6, 1, 81, 108, 54, 12, 1, 1024, 1280, 640, 160, 20, 1, 15625, 18750, 9375, 2500, 375, 30, 1, 279936, 326592, 163296, 45360, 7560, 756, 42, 1, 5764801, 6588344, 3294172, 941192, 168070, 19208, 1372, 56, 1, 134217728
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OFFSET
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0,5
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COMMENTS
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The same triangle (except for signs) may be obtained from the determinants of the Brahmagupta matrices, setting x->Sqrt[z], y->1, t->n. - Roger L. Bagula, Apr 09 2008
T(n,k)/A000312(n) is the probability P(n,k) that any member (j) of set J={1..n} will be selected k times given n random draws from J. This is equivalent to rolling an n-sided die (with standard assumptions) with sides numbered j=1..n: P(n,k) is the probability that any j will show k times with n rolls.
P(n,k) = (n-2)!*(n-1)^(n-k+1 )/k!*(n-k)!*n^(n-1); n>1. As n approaches infinity, P(n,0) and P(n,1) approach 1/e. (End)
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LINKS
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FORMULA
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T(n, k) = C(n, k)*(n-1)^(n-k), for n>1.
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EXAMPLE
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Triangle T(n,k) begins:
1;
0, 1;
1, 2, 1;
8, 12, 6, 1;
81, 108, 54, 12, 1;
1024, 1280, 640, 160, 20, 1;
15625, 18750, 9375, 2500, 375, 30, 1;
279936, 326592, 163296, 45360, 7560, 756, 42, 1;
...
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MATHEMATICA
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Clear[B] B[0] = {{x, y}, {t*y, x}}; B[n_] := B[n] = B[n - 1].B[0]; Table[Det[B[n]] /. x -> Sqrt[z] /. y -> 1 /. t -> n, {n, 0, 10}]; a = Join[{{1}}, Table[CoefficientList[Det[B[n]] /. x -> Sqrt[z] /. y ->1 /. t -> n, z], {n, 0, 10}]]; Flatten[a] (* Roger L. Bagula, Apr 09 2008 *)
row[n_] := CoefficientList[(x + n - 1)^n + O[x]^(n+1), x];
Join[{1, 0, 1}, Table[Binomial[n, k]*(n - 1)^(n - k), {n, 2, 49}, {k, 0, n}]] // Flatten (* G. C. Greubel, Nov 14 2017 *)
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PROG
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(PARI) for(n=0, 15, for(k=0, n, print1(if(n==0 && k==0, 1, if(n==1 && k==0, 0, if(n==1 && k==1, 1, binomial(n, k)*(n-1)^(n-k)))), ", "))) \\ G. C. Greubel, Nov 14 2017
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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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