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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. 9
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

Sequence in context: A026022 A073714 A171848 * A022818 A050447 A167172

Adjacent sequences:  A144148 A144149 A144150 * A144152 A144153 A144154

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 12 2008

STATUS

approved

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Last modified November 21 22:32 EST 2017. Contains 295054 sequences.