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A274938 Number of unlabeled forests with n nodes that have two components, neither of which is the empty graph.

%I A274938 #14 Mar 15 2017 09:03:52
%S A274938 0,0,0,1,1,3,5,11,21,46,96,216,482,1121,2633,6334,15414,38132,95321,
%T A274938 241029,614862,1582030,4099922,10697038,28074752,74086468,196469601,
%U A274938 523383136,1400048426,3759508536,10131089877,27391132238,74283533023,202030012554,550934011491,1506161266348
%N A274938 Number of unlabeled forests with n nodes that have two components, neither of which is the empty graph.
%H A274938 Alois P. Heinz, <a href="/A274938/b274938.txt">Table of n, a(n) for n = 0..1000</a>
%F A274938 G.f.: [A(x)^2 - A(x^2)]/2 where A(x) is the o.g.f. for A000055 without the initial constant 1.
%F A274938 a(2n+1) = A274937(2n+1). a(2n) = A274937(2n)-A000055(n). - _R. J. Mathar_, Jul 20 2016
%p A274938 with(numtheory):
%p A274938 b:= proc(n) option remember; `if`(n<2, n, (add(add(d*
%p A274938       b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
%p A274938     end:
%p A274938 g:= proc(n) option remember; `if`(n=0, 1, b(n)-add(b(j)*
%p A274938       b(n-j), j=0..n/2)+`if`(n::odd, 0, (t->t*(t+1)/2)(b(n/2))))
%p A274938     end:
%p A274938 a:= proc(n) option remember; add(g(j)*g(n-j), j=1..n/2)-
%p A274938       `if`(n::odd or n=0, 0, (t-> t*(t+1)/2)(g(n/2)))
%p A274938     end:
%p A274938 seq(a(n), n=0..40);  # _Alois P. Heinz_, Jul 20 2016
%t A274938 b[n_] := b[n] = If[n<2, n, Sum[DivisorSum[j, #*b[#]&]*b[n-j], {j, 1, n-1}]/(n-1)];
%t A274938 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 A274938 a[n_] := a[n] = Sum[g[j]*g[n-j], {j, 1, n/2}]-If[OddQ[n] || n==0, 0, Function[t, t*(t+1)/2][g[n/2]]];
%t A274938 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Mar 15 2017, after _Alois P. Heinz_ *)
%Y A274938 Cf. A000055, A274935-A274937. [A274935, A274936, A274937, A274938] are analogs for forests of [A275165, A275166, A216785, A274934] for graphs.
%K A274938 nonn
%O A274938 0,6
%A A274938 _N. J. A. Sloane_, Jul 19 2016