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 A185070 Triangular array read by rows.  T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} that have exactly k 3-cycles. n>=0, 0<=k<=floor(n/3). 0
 1, 1, 4, 25, 2, 224, 32, 2625, 500, 38056, 8560, 40, 657433, 164150, 1960, 13178880, 3526656, 71680, 300585601, 84389928, 2442720, 2240, 7683776000, 2232672000, 83328000, 224000, 217534555161, 64830707370, 2931500880, 14907200 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The total number of 3-cycles over all functions on {1,2,...,n} is 2*binomial(n,3)*n^(n-3).  So we see that as n gets large the probability that a random function would contain k 3-cycles is a Poisson distribution with mean = 1/3.  Generally, the total number of j-cycles over all functions on {1,2,...,n} is (j-1)!*binomial(n,j)*n^(n-j). LINKS FORMULA E.g.f.: exp(T(x)^3/3*(y - 1))/(1-T(x)) where T(x) is the e.g.f. for A000169. EXAMPLE 1, 1, 4, 25,        2, 224,       32, 2625,      500, 38056,     8560,     40, 657433,    164150,   1960, 13178880,  3526656,  71680, 300585601, 84389928, 2442720, 2240 MATHEMATICA nn=10; t=Sum[n^(n-1)x^n/n!, {n, 1, nn}]; Range[0, nn]!CoefficientList[Series[Exp[t^3/3(y-1)]/(1-t), {x, 0, nn}], {x, y}]//Grid CROSSREFS Cf. A185025, A055134 A190314 Sequence in context: A058230 A162187 A103644 * A272680 A082202 A283452 Adjacent sequences:  A185067 A185068 A185069 * A185071 A185072 A185073 KEYWORD nonn,tabf AUTHOR Geoffrey Critzer, Dec 25 2012 STATUS approved

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Last modified April 15 01:22 EDT 2021. Contains 342971 sequences. (Running on oeis4.)