A144151 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) = number of ways an undirected cycle of length k can be built from n labeled nodes.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 3, 1, 5, 10, 10, 15, 12, 1, 6, 15, 20, 45, 72, 60, 1, 7, 21, 35, 105, 252, 420, 360, 1, 8, 28, 56, 210, 672, 1680, 2880, 2520, 1, 9, 36, 84, 378, 1512, 5040, 12960, 22680, 20160, 1, 10, 45, 120, 630, 3024, 12600, 43200, 113400, 201600, 181440

Offset

0,5

Links

Alois P. Heinz, Rows n = 0..140, flattened

Formula

T(n,k) = C(n,k) if k<=2, else T(n,k) = C(n,k)*(k-1)!/2.

E.g.f.: exp(x)*(log(1/(1 - y*x))/2 + 1 + y*x/2 + (y*x)^2/4). - Geoffrey Critzer, Jul 22 2016

Example

T(4,3) = 4, because 4 undirected cycles of length 3 can be built from 4 labeled nodes:
  .1.2. .1.2. .1-2. .1-2.
  ../|. .|\.. ..\|. .|/..
  .3-4. .3-4. .3.4. .3.4.
Triangle begins:
  1;
  1, 1;
  1, 2,  1;
  1, 3,  3,  1;
  1, 4,  6,  4,  3;
  1, 5, 10, 10, 15, 12;
  ...

Maple

T:= (n, k)-> if k<=2 then binomial(n, k) else mul(n-j, j=0..k-1)/k/2 fi:
seq(seq(T(n, k), k=0..n), n=0..12);

Mathematica

t[n_, k_ /; k <= 2] := Binomial[n, k]; t[n_, k_] := Binomial[n, k]*(k-1)!/2; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 18 2013 *)
CoefficientList[Table[1 + n x (2 + (n - 1) x + 2 HypergeometricPFQ[{1, 1, 1 - n}, {2}, -x])/4, {n, 0, 10}], x] (* Eric W. Weisstein, Apr 06 2017 *)

Crossrefs

Columns 0-4 give: A000012, A000027, A000217, A000292, A050534.

Diagonal gives: A001710.

Cf. A000142, A007318.

Row sums are in A116723. - Alois P. Heinz, Jun 01 2009

Excluding columns k=0,1,and 2 the row sums are A002807. - Geoffrey Critzer, Jul 22 2016

Cf. A284947 (k-cycle counts for k >= 3 in the complete graph K_n). - Eric W. Weisstein, Apr 06 2017

T(2n,n) gives A006963(n+1) for n>=3.

Sequence in context: A026022 A073714 A171848 * A022818 A050447 A248601

Adjacent sequences: A144148 A144149 A144150 * A144152 A144153 A144154

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 12 2008

Status

approved