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A038055 Number of n-node rooted trees with nodes of 2 colors.

%I A038055 #45 Dec 26 2020 10:01:03
%S A038055 2,4,14,52,214,916,4116,18996,89894,433196,2119904,10503612,52594476,
%T A038055 265713532,1352796790,6933598208,35747017596,185260197772,
%U A038055 964585369012,5043220350012,26467146038744,139375369621960,736229024863276,3900074570513316,20714056652990194
%N A038055 Number of n-node rooted trees with nodes of 2 colors.
%H A038055 Alois P. Heinz, <a href="/A038055/b038055.txt">Table of n, a(n) for n = 1..1000</a>
%H A038055 L. Foissy, <a href="https://arxiv.org/abs/1811.07572">Algebraic structures on typed decorated rooted trees</a>, arXiv:1811.07572 [math.RA] (2018).
%H A038055 R. J. Mathar, <a href="http://arxiv.org/abs/1603.00077">Topologically distinct sets of non-intersecting circles in the plane</a>, arXiv:1603.00077 [math.CO] (2016), Table 3.
%H A038055 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A038055 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H A038055 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A038055 Shifts left and halves under Euler transform.
%F A038055 a(n) = 2*A000151(n).
%F A038055 a(n) ~ c * d^n / n^(3/2), where d = A245870 = 5.646542616232949712892713516..., c = 0.41572319484583484264330698410170337587070758092051645875080558178621559423... . - _Vaclav Kotesovec_, Sep 11 2014, updated Dec 26 2020
%p A038055 spec := [N, {N=Prod(bead,Set(N)), bead=Union(R,B), R=Atom, B=Atom}]; [seq(combstruct[count](spec, size=n), n=1..40)];
%p A038055 # second Maple program:
%p A038055 with(numtheory):
%p A038055 a:= proc(n) option remember; `if`(n<2, 2*n, (add(add(d*
%p A038055       a(d), d=divisors(j))*a(n-j), j=1..n-1))/(n-1))
%p A038055     end:
%p A038055 seq(a(n), n=1..30);  # _Alois P. Heinz_, May 11 2014
%t A038055 a[n_] := a[n] = If[n<2, 2*n, (Sum[Sum[d*a[d], {d, Divisors[j]}]*a[n-j], {j, 1, n-1}])/(n-1)]; Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Feb 25 2015, after _Alois P. Heinz_ *)
%t A038055 a[1] = 2; a[n_] := a[n] = Sum[k a[k] a[n - m k]/(n-1), {k, n}, {m, (n-1)/k}]; Table[a[n], {n, 30}] (* _Vladimir Reshetnikov_, Aug 12 2016 *)
%o A038055 (PARI) seq(N) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 2/n * sum(i=1, n, sumdiv(i, d, d*A[d]) * A[n-i+1] ) ); 2*A} \\ _Andrew Howroyd_, May 12 2018
%Y A038055 Cf. A000081, A038056-A038062, A271878 (multisets).
%Y A038055 Cf. A245870.
%K A038055 nonn,eigen,nice,easy
%O A038055 1,1
%A A038055 _Christian G. Bower_, Jan 04 1999