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A248869 Satisfies Sum_{n>=0} a(n)*x^n = x * Product_{n>=0} (1 + x^n + x^(2*n))^a(n).

%I A248869 #56 Oct 02 2019 22:47:13
%S A248869 0,1,1,2,3,7,15,34,79,190,459,1136,2833,7154,18206,46723,120656,
%T A248869 313514,818763,2148434,5660790,14972103,39734107,105779291,282403830,
%U A248869 755921733,2028277115,5454368549,14697955778,39682793675,107330573239,290783511134,789032648219
%N A248869 Satisfies Sum_{n>=0} a(n)*x^n = x * Product_{n>=0} (1 + x^n + x^(2*n))^a(n).
%C A248869 What kind of trees are counted by this sequence (compare with A000081, A004111, A073075, and A115593)?
%C A248869 a(n) is the number of rooted trees of n vertices that have everywhere at most 2 siblings with the same (i.e., isomorphic) subtree below.  The g.f. assembles a(n) as a root with child subtrees from among the smaller a(), but takes only 0, 1 or 2 copies of any one of them.  Compare asymmetric trees A004111 g.f. which takes 0 or 1 copies.  Here the x^(2*n) term allows a 2nd copy.  The siblings condition is equivalent to the condition that the tree automorphisms form a 2-group, i.e., group order some power 2^k.  2 same siblings are a swap.  3 same siblings would be an element of order 3 and hence factor 3 in the group order.  a(n) >= A213920 since the latter limits same size siblings, whereas here only limits same size plus structure. - _Kevin Ryde_, Jul 11 2019
%H A248869 Alois P. Heinz, <a href="/A248869/b248869.txt">Table of n, a(n) for n = 0..2213</a>
%F A248869 a(n) ~ c * d^n / n^(3/2), where d = 2.8458470164106425911151048..., c = 0.41694347809945986693376... . - _Vaclav Kotesovec_, Mar 17 2015
%F A248869 a(n) = A004111(n) + A318859(n). - _Kevin Ryde_, Jul 11 2019
%p A248869 h:= proc(n, m, t) option remember; `if`(m=0, binomial(n+t, t),
%p A248869       `if`(n=0, 0, add(h(n-1, m-j, t+1), j=1..min(2, m))))
%p A248869     end:
%p A248869 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A248869       add(b(n-i*j, i-1)*h(a(i), j, 0), j=0..n/i)))
%p A248869     end:
%p A248869 a:= n-> `if`(n<2, n, b(n-1$2)):
%p A248869 seq(a(n), n=0..35);  # _Alois P. Heinz_, Sep 04 2018
%t A248869 h[n_, m_, t_] := h[n, m, t] = If[m == 0, Binomial[n + t, t], If[n == 0, 0, Sum[h[n - 1, m - j, t + 1], {j, 1, Min[2, m]}]]];
%t A248869 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, i - 1]* h[a[i], j, 0], {j, 0, n/i}]]];
%t A248869 a[n_] := If[n < 2, n, b[n - 1, n - 1]];
%t A248869 a /@ Range[0, 32] (* _Jean-François Alcover_, Oct 02 2019, after _Alois P. Heinz_ *)
%Y A248869 Cf. A000081, A004111, A073075, A115593, A213920, A248870, A309352.
%Y A248869 Column k=2 of A318757.
%K A248869 nonn
%O A248869 0,4
%A A248869 _Joerg Arndt_, Mar 04 2015