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A201013 Triangular array read by rows: T(n,k) is the number of 2-regular labeled graphs on n nodes that have exactly k connected components (cycles); n>=3, 1<=k<=floor(n/3).

%I A201013 #25 Nov 14 2014 10:48:06
%S A201013 1,3,12,60,10,360,105,2520,987,20160,9576,280,181440,99144,6300,
%T A201013 1814400,1104840,107415,19958400,13262040,1708245,15400,239500800,
%U A201013 171119520,27042444,600600,3113510400,2366076960,437729292,16186170,43589145600,34941291840,7335055728,382056675,1401400
%N A201013 Triangular array read by rows: T(n,k) is the number of 2-regular labeled graphs on n nodes that have exactly k connected components (cycles); n>=3, 1<=k<=floor(n/3).
%C A201013 A 2-regular labeled graph is a simple labeled graph such that every vertex has degree 2.
%H A201013 Alois P. Heinz, <a href="/A201013/b201013.txt">Rows n = 3..170, flattened</a>
%F A201013 E.g.f.: exp(-xy/2-x^2y/4)/(1-x)^(y/2).
%F A201013 T(n,1) = (n-1)!/2, T(n,k) = Sum_{j=3..n-3} C(n-1,j-1)*T(j,1)*T(n-j,k-1) for k>1. - _Alois P. Heinz_, Nov 25 2011
%F A201013 Sum_{k=1..floor(n/3)} T(n,k)*2^k = A038205(n) the number of permutations with minimum cycle size of 3. - _Geoffrey Critzer_, Nov 05 2012
%e A201013 1;
%e A201013 3;
%e A201013 12;
%e A201013 60,     10;
%e A201013 360,    105;
%e A201013 2520,   987;
%e A201013 20160,  9576,    280;
%e A201013 181440, 99144,   6300;
%p A201013 T:= proc(n, k) option remember; `if`(k=1, (n-1)!/2,
%p A201013       add(binomial(n-1, j-1) *T(j,1) *T(n-j, k-1), j=3..n-3))
%p A201013     end:
%p A201013 seq(seq(T(n, k), k=1..n/3), n=3..14); # _Alois P. Heinz_, Nov 25 2011
%t A201013 f[list_]:=Select[list,#>0&];Flatten[Drop[Map[f, a = Log[1/(1 - x)]/2 - x/2 - x^2/4; Range[0, 20]! CoefficientList[Series[Exp[y a], {x, 0, 20}], {x, y}]], 3]]
%Y A201013 Cf. A001205 (row sums), A001710(n-1) (first row).
%K A201013 nonn,tabf
%O A201013 3,2
%A A201013 _Geoffrey Critzer_, Nov 25 2011