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A244457 Number of unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 3.

%I A244457 #13 Feb 06 2015 08:11:56
%S A244457 1,0,0,1,2,2,4,7,12,20,34,56,98,167,284,484,835,1433,2467,4250,7345,
%T A244457 12700,22004,38154,66266,115224,200623,349654,610126,1065739,1863547,
%U A244457 3261672,5714277,10020092,17586014,30890654,54305289,95542387,168221056,296401979
%N A244457 Number of unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 3.
%H A244457 Alois P. Heinz, <a href="/A244457/b244457.txt">Table of n, a(n) for n = 4..1000</a>
%F A244457 a(n) ~ c * d^n / n^(3/2), where d = 1.8239199077079..., c = 0.49573400799... . - _Vaclav Kotesovec_, Jul 11 2014
%e A244457 a(7) = 1:
%e A244457       o
%e A244457      /|\
%e A244457     o o o
%e A244457    /|\
%e A244457   o o o
%p A244457 b:= proc(n, i, t, k) option remember; `if`(n=0, `if`(t in [0, k],
%p A244457       1, 0), `if`(i<1 or t>n, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
%p A244457       b(n-i*j, i-1, max(0,t-j), k), j=0..n/i)))
%p A244457     end:
%p A244457 a:= n-> b(n-1$2, 3$2) -b(n-1$2, 4$2):
%p A244457 seq(a(n), n=4..45);
%t A244457 b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[i < 1, 0, Sum[Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, Max[0, t - j], k], {j, 0, n/i}]] // FullSimplify]; a[n_] := b[n - 1, n - 1, 3, 3] - b[n - 1, n - 1, 4, 4]; Table[a[n], {n, 4, 45}] (* _Jean-François Alcover_, Feb 06 2015, after Maple *)
%Y A244457 Column k=3 of A244454.
%Y A244457 Cf. A244532.
%K A244457 nonn
%O A244457 4,5
%A A244457 _Joerg Arndt_ and _Alois P. Heinz_, Jun 29 2014