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Number of simple unlabeled graphs on n nodes with exactly 4 connected components that are trees or cycles.
3

%I #8 Dec 04 2014 06:54:29

%S 1,1,3,6,13,25,53,106,224,475,1037,2315,5302,12382,29495,71450,175618,

%T 437272,1101000,2799768,7181703,18565546,48326482,126581749,333424119,

%U 882765486,2348114866,6272676477,16822741588,45281587081,122295734356,331331765694

%N Number of simple unlabeled graphs on n nodes with exactly 4 connected components that are trees or cycles.

%H Alois P. Heinz, <a href="/A215984/b215984.txt">Table of n, a(n) for n = 4..650</a>

%e a(6) = 3: .o-o o. .o-o o. .o o o.

%e .|/ . .| . .| | .

%e .o o o. .o o o. .o o o.

%p with(numtheory):

%p b:= proc(n) option remember; local d, j; `if`(n<=1, n,

%p (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/(n-1))

%p end:

%p g:= proc(n) option remember; local k; `if`(n>2, 1, 0)+ b(n)-

%p (add(b(k)*b(n-k), k=0..n) -`if`(irem(n, 2)=0, b(n/2), 0))/2

%p end:

%p p:= proc(n, i, t) option remember; `if`(n<t, 0, `if`(n=t, 1,

%p `if`(min(i, t)<1, 0, add(binomial(g(i)+j-1, j)*

%p p(n-i*j, i-1, t-j), j=0..min(n/i,t)))))

%p end:

%p a:= n-> p(n, n, 4):

%p seq(a(n), n=4..40);

%Y Column k=4 of A215977.

%Y The labeled version is A215854.

%K nonn

%O 4,3

%A _Alois P. Heinz_, Aug 29 2012