A201013 - OEIS

%I #25 Nov 14 2014 10:48:06

%S 1,3,12,60,10,360,105,2520,987,20160,9576,280,181440,99144,6300,

%T 1814400,1104840,107415,19958400,13262040,1708245,15400,239500800,

%U 171119520,27042444,600600,3113510400,2366076960,437729292,16186170,43589145600,34941291840,7335055728,382056675,1401400

%N 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 A 2-regular labeled graph is a simple labeled graph such that every vertex has degree 2.

%H Alois P. Heinz, <a href="/A201013/b201013.txt">Rows n = 3..170, flattened</a>

%F E.g.f.: exp(-xy/2-x^2y/4)/(1-x)^(y/2).

%F 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 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 1;

%e 3;

%e 12;

%e 60,     10;

%e 360,    105;

%e 2520,   987;

%e 20160,  9576,    280;

%e 181440, 99144,   6300;

%p T:= proc(n, k) option remember; `if`(k=1, (n-1)!/2,

%p       add(binomial(n-1, j-1) *T(j,1) *T(n-j, k-1), j=3..n-3))

%p     end:

%p seq(seq(T(n, k), k=1..n/3), n=3..14); # _Alois P. Heinz_, Nov 25 2011

%t 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 Cf. A001205 (row sums), A001710(n-1) (first row).

%K nonn,tabf

%O 3,2

%A _Geoffrey Critzer_, Nov 25 2011