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A144212 Triangle T(n,k), n>=3, 3<=k<=n, read by rows: T(n,k) = number of simple graphs on n labeled nodes, where each maximally connected subgraph consists of a single node or has a unique cycle of length k.

%I A144212 #15 Feb 28 2015 22:39:47
%S A144212 2,17,4,221,76,13,3261,1486,433,61,54801,29506,11593,2941,361,1049235,
%T A144212 628531,296353,102481,23041,2521,22695027,14633011,7795873,3270961,
%U A144212 1010881,204121,20161,548904831,373486051,217126225,104038201,39355201
%N A144212 Triangle T(n,k), n>=3, 3<=k<=n, read by rows: T(n,k) = number of simple graphs on n labeled nodes, where each maximally connected subgraph consists of a single node or has a unique cycle of length k.
%H A144212 Alois P. Heinz, <a href="/A144212/b144212.txt">Rows n = 3..102, flattened</a>
%F A144212 See program.
%e A144212 T(4,4) = 4, because there are 4 simple graphs on 4 labeled nodes, where each maximally connected subgraph consists of a single node or has a unique cycle of length 4:
%e A144212 .1.2. .1-2. .1-2. .1.2.
%e A144212 ..... .|.|. ..X.. .|X|.
%e A144212 .3.4. .3-4. .3-4. .3.4.
%e A144212 Triangle begins:
%e A144212         2;
%e A144212        17,     4;
%e A144212       221,    76,    13;
%e A144212      3261,  1486,   433,   61;
%e A144212     54801, 29506, 11593, 2941, 361;
%p A144212 B:= proc(n,c,k) option remember; if c=0 then 1 elif c<0 or n<c then 0 elif c=n then binomial(n-1,k-1) *(k-1)! /2 *n^(n-k) else B(n-1,c,k) +add(binomial(n-1,j) * B(j+1,j+1,k) *B(n-1-j,c-j-1,k), j=k-1..c-1) fi end: T:= (n,k)-> add(B(n,c,k), c=0..n): seq(seq(T(n,k), k=3..n), n=3..11);
%t A144212 B[n_, c_, k_] := B[n, c, k] = Which[c == 0, 1, c<0 || n<c, 0, c == n, Binomial[n-1, k-1]*(k-1)!/2*n^(n-k), True, B[n-1, c, k] + Sum[Binomial[n-1, j]*B[j+1, j+1, k]*B[n-1-j, c-j-1, k], {j, k-1, c-1}]]; T[n_, k_] := Sum [B[n, c, k], {c, 0, n}]; Table[Table [T[n, k], {k, 3, n}], {n, 3, 11}] // Flatten (* _Jean-François Alcover_, Jan 21 2014, translated from _Alois P. Heinz_'s Maple code *)
%Y A144212 Columns k=3, 4 give: A144208, A144210. Diagonal gives: A139149. Cf. A053507, A065889, A098909, A144207, A144209, A007318, A000142.
%K A144212 nonn,tabl,look
%O A144212 3,1
%A A144212 _Alois P. Heinz_, Sep 14 2008