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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).

%I A143899 #8 Feb 14 2014 08:57:46
%S A143899 1,4,15,6,1,10,85,252,210,120,45,10,1,20,285,1707,5005,6435,6435,5005,
%T A143899 3003,1365,455,105,15,1,35,735,6972,37457,116280,203490,293930,352716,
%U A143899 352716,293930,203490,116280,54264,20349,5985,1330,210,21,1,56,1610
%N 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).
%H A143899 Alois P. Heinz, <a href="/A143899/b143899.txt">Table of n, a(n) for n = 3..10585</a>
%F A143899 T(n,k) = A084546(n,k)-A138464(n,k).
%e A143899 T(4,3) = 4, because 4 simple graphs on 4 labeled nodes with 3 edges contain a cycle subgraph:
%e A143899 ..1-2...1-2...1.2...1.2..
%e A143899 ..|/.....\|...|\...../|..
%e A143899 ..3.4...3.4...3-4...3-4..
%e A143899 Triangle begins:
%e A143899 1;
%e A143899 4,   15,    6,    1;
%e A143899 10,  85,  252,  210,  120,   45,   10,    1;
%e A143899 20, 285, 1707, 5005, 6435, 6435, 5005, 3003, 1365, 455, 105, 15, 1;
%p A143899 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);
%t A143899 (* 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 *)
%Y A143899 Row sums give A143900. Cf. A084546, A138464, A007318.
%K A143899 nonn,tabf
%O A143899 3,2
%A A143899 _Alois P. Heinz_, Sep 04 2008