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A215983 Number of simple unlabeled graphs on n nodes with exactly 3 connected components that are trees or cycles.

%I A215983 #15 Dec 18 2020 11:55:55
%S A215983 1,1,3,6,12,23,47,92,189,401,869,1949,4475,10520,25183,61366,151555,
%T A215983 379164,958555,2446746,6296819,16326996,42613240,111889355,295372835,
%U A215983 783598713,2088175182,5587741350,15009229137,40458659246,109416872688,296810505298
%N A215983 Number of simple unlabeled graphs on n nodes with exactly 3 connected components that are trees or cycles.
%H A215983 Alois P. Heinz, <a href="/A215983/b215983.txt">Table of n, a(n) for n = 3..650</a>
%F A215983 a(n) ~ c * d^n / n^(5/2), where d = A051491 = 2.9557652856519949747148..., c = 0.13932434077355395... . - _Vaclav Kotesovec_, Sep 07 2014
%e A215983 a(5) = 3: .o o-o.  .o o-o.  .o o o.
%e A215983           .  |/ .  .  |  .  .| |  .
%e A215983           .o o  .  .o o  .  .o o  .
%p A215983 with(numtheory):
%p A215983 b:= proc(n) option remember; local d, j; `if`(n<=1, n,
%p A215983       (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/(n-1))
%p A215983     end:
%p A215983 g:= proc(n) option remember; local k; `if`(n>2, 1, 0)+ b(n)-
%p A215983       (add(b(k)*b(n-k), k=0..n) -`if`(irem(n, 2)=0, b(n/2), 0))/2
%p A215983     end:
%p A215983 p:= proc(n, i, t) option remember; `if`(n<t, 0, `if`(n=t, 1,
%p A215983       `if`(min(i, t)<1, 0, add(binomial(g(i)+j-1, j)*
%p A215983        p(n-i*j, i-1, t-j), j=0..min(n/i,t)))))
%p A215983     end:
%p A215983 a:= n-> p(n, n, 3):
%p A215983 seq(a(n), n=3..40);
%t A215983 b[n_] := b[n] = If[n <= 1, n, Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n - j], {j, 1, n - 1}]/(n - 1)];
%t A215983 g[n_] := g[n] = If[n > 2, 1, 0] + b[n] - (Sum[b[k]*b[n - k], {k, 0, n}] - If[Mod[n, 2] == 0, b[n/2], 0])/2;
%t A215983 p[n_, i_, t_] := p[n, i, t] = If[n < t, 0, If[n == t, 1, If[Min[i, t] < 1, 0, Sum[Binomial[g[i] + j - 1, j]*p[n - i*j, i - 1, t - j], {j, 0, Min[n/i, t]}]]]];
%t A215983 a[n_] := p[n, n, 3];
%t A215983 a /@ Range[3, 40] (* _Jean-François Alcover_, Dec 18 2020, after _Alois P. Heinz_ *)
%Y A215983 Column k=3 of A215977.
%Y A215983 The labeled version is A215853.
%K A215983 nonn
%O A215983 3,3
%A A215983 _Alois P. Heinz_, Aug 29 2012