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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).

%I A185070 #23 Dec 15 2021 10:25:01
%S A185070 1,1,4,25,2,224,32,2625,500,38056,8560,40,657433,164150,1960,13178880,
%T A185070 3526656,71680,300585601,84389928,2442720,2240,7683776000,2232672000,
%U A185070 83328000,224000,217534555161,64830707370,2931500880,14907200
%N 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).
%C A185070 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).
%F A185070 E.g.f.: exp(T(x)^3/3*(y - 1))/(1-T(x)) where T(x) is the e.g.f. for A000169.
%e A185070           1;
%e A185070           1;
%e A185070           4;
%e A185070          25,        2;
%e A185070         224,       32;
%e A185070        2625,      500;
%e A185070       38056,     8560,      40;
%e A185070      657433,   164150,    1960;
%e A185070    13178880,  3526656,   71680;
%e A185070   300585601, 84389928, 2442720, 2240;
%e A185070   ...
%t A185070 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
%Y A185070 Cf. A185025, A055134, A190314.
%K A185070 nonn,tabf
%O A185070 0,3
%A A185070 _Geoffrey Critzer_, Dec 25 2012