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

%I #13 Feb 06 2015 09:15:14

%S 1,0,0,3,7,8,21,55,121,265,611,1379,3193,7436,17085,39339,91846,

%T 214549,500132,1169267,2743302,6445797,15167805,35749961,84390645,

%U 199523566,472429633,1120012481,2658525869,6318368820,15034189965,35811690663,85393261630

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

%H Alois P. Heinz, <a href="/A244532/b244532.txt">Table of n, a(n) for n = 4..1000</a>

%H Vaclav Kotesovec, <a href="/A244532/a244532.txt">Recurrence (of order 9)</a>

%F a(n) ~ sqrt(sqrt(2)/4 - sqrt(154+112*sqrt(2))/56) * ((sqrt(13+16*sqrt(2))-1)/2)^n / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jul 02 2014

%p b:= proc(n, t, k) option remember; `if`(n=0,

%p `if`(t in [0, k], 1, 0), `if`(t>n, 0, add(b(j-1, k$2)*

%p b(n-j, max(0, t-1), k), j=1..n)))

%p end:

%p a:= n-> b(n-1, 3$2) -b(n-1, 4$2):

%p seq(a(n), n=4..40);

%t b[n_, t_, k_] := b[n, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[t > n, 0, Sum[b[j - 1, k, k]*b[n - j, Max[0, t - 1], k], {j, 1, n}]]]; T[n_, k_] := b[n - 1, k, k] - If[n == 1 && k == 0, 0, b[n - 1, k + 1, k + 1]]; a[n_] := b[n - 1, 3, 3] - b[n - 1, 4, 4]; Table[a[n], {n, 4, 40}] (* _Jean-François Alcover_, Feb 06 2015, after Maple *)

%Y Column k=3 of A244530.

%Y Cf. A244457.

%K nonn

%O 4,4

%A _Joerg Arndt_ and _Alois P. Heinz_, Jun 29 2014