A143899 Triangle read by rows: T(n,k)=number of simple graphs on n labeled nodes with k edges containing at least one cycle subgraph, n>=3, 3<=k<=C(n,2).

1, 4, 15, 6, 1, 10, 85, 252, 210, 120, 45, 10, 1, 20, 285, 1707, 5005, 6435, 6435, 5005, 3003, 1365, 455, 105, 15, 1, 35, 735, 6972, 37457, 116280, 203490, 293930, 352716, 352716, 293930, 203490, 116280, 54264, 20349, 5985, 1330, 210, 21, 1, 56, 1610

Offset: 3

Author

Alois P. Heinz, Sep 04 2008

Keywords

• nonn
• tabf

Examples

			T(4,3) = 4, because 4 simple graphs on 4 labeled nodes with 3 edges contain a cycle subgraph:
..1-2...1-2...1.2...1.2..
..|/.....\|...|\...../|..
..3.4...3.4...3-4...3-4..
Triangle begins:
1;
4,   15,    6,    1;
10,  85,  252,  210,  120,   45,   10,    1;
20, 285, 1707, 5005, 6435, 6435, 5005, 3003, 1365, 455, 105, 15, 1;
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..10585

Crossrefs

Row sums give A143900. Cf. A084546, A138464, A007318.

Programs

Maple

B:= proc(n) option remember; if n=0 then 0 else B(n-1) +n^(n-1) *x^n/n! fi end: BB:= proc(n) option remember; expand (B(n) -B(n)^2/2) end: f:= proc(k) option remember; if k=0 then 1 else unapply (f(k-1)(x) +x^k/k!, x) fi end: A:= proc(n,k) option remember; series(f(k)(BB(n)), x,n+1) end: aa:= (n,k)-> coeff (A(n,k), x,n) *n!: b:= (n,k)-> if k>=n then 0 else aa(n,n-k) -aa(n,n-k-1) fi: T:= (n,k)-> product (n*(n-1)/2-j, j=0..k-1)/k! -b(n,k): seq (seq (T(n,k), k=3..n*(n-1)/2), n=3..8);

Mathematica

(* t = A138464 *) t[0, 0] = 1; t[n_, k_] /; (0 <= k <= n-1) := t[n, k] = Sum[(i+1)^(i-1)*Binomial[n-1, i]*t[n-i-1, k-i], {i, 0, k}]; t[, ] = 0; T[n_, k_] := Binomial[n*(n-1)/2, k]-t[n, k]; Table[Table[T[n, k], {k, 3, n*(n-1)/2}], {n, 3, 8}] // Flatten (* Jean-François Alcover, Feb 14 2014 *)

Formula

T(n,k) = A084546(n,k)-A138464(n,k).