|
%I #15 Dec 16 2014 14:02:43
%S 1,1,1,1,4,76,1486,29506,628531,14633011,373486051,10423892971,
%T 316702467496,10422938835196,369779598658786,14078057663869606,
%U 572776958092098166,24810200300393961286,1140218754844983978646
%N 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 4; also row sums of A144209.
%H Alois P. Heinz, <a href="/A144210/b144210.txt">Table of n, a(n) for n = 0..150</a>
%F a(n) = Sum_{k=0..n} A144209(n,k).
%F a(n) ~ c * n^(n-1), where c = 0.7519160836660874254... . - _Vaclav Kotesovec_, Sep 10 2014
%e a(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 .1.2. .1-2. .1-2. .1.2.
%e ..... .|.|. ..X.. .|X|.
%e .3.4. .3-4. .3-4. .3.4.
%p T:= proc(n,k) option remember; if k=0 then 1 elif k<0 or n<k then 0 elif k=n then 3*binomial(n-1,3)*n^(n-4) else T(n-1,k) +add(binomial(n-1,j) * T(j+1,j+1) *T(n-1-j,k-j-1), j=3..k-1) fi end: a:= n-> add(T(n,k), k=0..n): seq(a(n), n=0..23);
%t T[n_, k_] := T[n, k] = Which[k == 0, 1, k<0 || n<k, 0, k == n, 3*Binomial[n-1, 3]*n^(n-4), True, T[n-1, k] + Sum[Binomial[n-1, j]*T[j+1, j+1]*T[n-1-j, k-j-1], {j, 3, k-1}]]; a[n_] := Sum[T[n, k], {k, 0, n}]; Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Dec 02 2014, translated from Maple *)
%Y A column of A144212. Cf. A144209.
%K nonn
%O 0,5
%A _Alois P. Heinz_, Sep 14 2008
|
