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A185025 Triangular array read by rows: T(n,k) is the number of functions f:{1,2,...,n} -> {1,2,...,n} that have exactly k 2-cycles for n >= 0 and 0 <= k <= floor(n/2).

%I A185025 #20 Jan 21 2023 09:08:00
%S A185025 1,1,3,1,18,9,163,90,3,1950,1100,75,28821,16245,1575,15,505876,283122,
%T A185025 33810,735,10270569,5699932,780150,26460,105,236644092,130267440,
%U A185025 19615932,884520,8505,6098971555,3332614725,538325550,29619450,467775,945
%N A185025 Triangular array read by rows: T(n,k) is the number of functions f:{1,2,...,n} -> {1,2,...,n} that have exactly k 2-cycles for n >= 0 and 0 <= k <= floor(n/2).
%C A185025 It appears that as n gets large, row n conforms to a Poisson distribution with mean = 1/2.  In other words, as n gets large, T(n,k) approaches n^n/(2^k*k!*e^(1/2)).
%F A185025 E.g.f.: exp((T(x)^2/2)*(y-1))/(1 - T(x)) where T(x) is the e.g.f. for A000169.
%F A185025 Sum_{k=1..floor(n/2)} k * T(n,k) = A081131(n).
%e A185025 Triangle begins:
%e A185025            1;
%e A185025            1;
%e A185025            3,          1;
%e A185025           18,          9;
%e A185025          163,         90,         3;
%e A185025         1950,       1100,        75;
%e A185025        28821,      16245,      1575,       15;
%e A185025       505876,     283122,     33810,      735;
%e A185025     10270569,    5699932,    780150,    26460,    105;
%e A185025    236644092,  130267440,  19615932,   884520,   8505;
%e A185025   6098971555, 3332614725, 538325550, 29619450, 467775, 945;
%e A185025   ...
%t A185025 nn=10;t=Sum[n^(n-1)x^n/n!,{n,1,nn}]; Range[0,nn]! CoefficientList[Series[Exp[t^2/2(y-1)]/(1-t), {x,0,nn}], {x,y}]//Grid
%Y A185025 Column k=0 gives A089466.
%Y A185025 Cf. A000169, A081131.
%K A185025 nonn,tabf
%O A185025 0,3
%A A185025 _Geoffrey Critzer_, Dec 24 2012