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A274936 Number of n-node unlabeled forests that have 2 non-isomorphic components.

%I A274936 #20 Mar 15 2017 08:52:46
%S A274936 0,1,1,2,3,6,11,22,44,93,202,451,1033,2422,5792,14075,34734,86761,
%T A274936 219188,558984,1437927,3726535,9723678,25525112,67374649,178723358,
%U A274936 476263051,1274448596,3423491458,9229075121,24961961679,67721961268,184255943244,502658875034,1374713643212
%N A274936 Number of n-node unlabeled forests that have 2 non-isomorphic components.
%H A274936 Alois P. Heinz, <a href="/A274936/b274936.txt">Table of n, a(n) for n = 0..1000</a>
%F A274936 G.f.: [A(x)^2 - A(x^2)]/2 where A(x) is the o.g.f. for A000055.
%F A274936 a(2n+1) = A274935(2n+1). a(2n) = A274935(2n)-A000055(n). - _R. J. Mathar_, Jul 20 2016
%p A274936 with(numtheory):
%p A274936 b:= proc(n) option remember; `if`(n<2, n, (add(add(d*
%p A274936       b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
%p A274936     end:
%p A274936 g:= proc(n) option remember; `if`(n=0, 1, b(n)-add(b(j)*
%p A274936       b(n-j), j=0..n/2)+`if`(n::odd, 0, (t->t*(t+1)/2)(b(n/2))))
%p A274936     end:
%p A274936 a:= proc(n) option remember; add(g(j)*g(n-j), j=0..n/2)-
%p A274936       `if`(n::odd, 0, (t-> t*(t+1)/2)(g(n/2)))
%p A274936     end:
%p A274936 seq(a(n), n=0..40);  # _Alois P. Heinz_, Jul 20 2016
%t A274936 b[n_] := b[n] = If[n<2, n, Sum[DivisorSum[j, #*b[#]&]*b[n-j], {j, 1, n-1}]/(n-1)];
%t A274936 g[n_] := g[n] = If[n==0, 1, b[n]-Sum[b[j]*b[n-j], {j, 0, n/2}]+If[OddQ[n], 0, Function[t, t*(t+1)/2][b[n/2]]]];
%t A274936 a[n_] := a[n] = Sum[g[j]*g[n-j], {j, 0, n/2}]-If[OddQ[n], 0, Function[t, t*(t+1)/2][g[n/2]]];
%t A274936 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Mar 15 2017, after _Alois P. Heinz_ *)
%Y A274936 Cf. A000055, A274935-A274938. [A274935, A274936, A274937, A274938] are analogs for forests of [A275165, A275166, A216785, A274934] for graphs.
%K A274936 nonn
%O A274936 0,4
%A A274936 _N. J. A. Sloane_, Jul 19 2016