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).

1, 3, 12, 60, 10, 360, 105, 2520, 987, 20160, 9576, 280, 181440, 99144, 6300, 1814400, 1104840, 107415, 19958400, 13262040, 1708245, 15400, 239500800, 171119520, 27042444, 600600, 3113510400, 2366076960, 437729292, 16186170, 43589145600, 34941291840, 7335055728, 382056675, 1401400

Offset: 3

Author

Geoffrey Critzer, Nov 25 2011

Keywords

• nonn
• tabf

Comments

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

Examples

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

Links

• Alois P. Heinz, Rows n = 3..170, flattened

Crossrefs

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

Programs

Maple

T:= proc(n, k) option remember; `if`(k=1, (n-1)!/2,
      add(binomial(n-1, j-1) *T(j,1) *T(n-j, k-1), j=3..n-3))
    end:
seq(seq(T(n, k), k=1..n/3), n=3..14); # Alois P. Heinz, Nov 25 2011

Mathematica

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]]

Formula

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

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

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