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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

Table of n, a(n) for n=0..29.

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 January 19 06:37 EST 2020. Contains 331033 sequences. (Running on oeis4.)