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
Diagonal gives: A001710.
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.
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 12 2008
STATUS
approved