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A131892 a(n) is the number of shapes of balanced trees with constant branching factor 6 and n nodes. The node is balanced if the size, measured in nodes, of each pair of its children differ by at most one node.

%I A131892 #21 Aug 01 2019 20:30:06
%S A131892 1,1,6,15,20,15,6,1,36,540,4320,19440,46656,46656,699840,4374000,
%T A131892 14580000,27337500,27337500,11390625,91125000,303750000,540000000,
%U A131892 540000000,288000000,64000000,288000000,540000000,540000000,303750000,91125000,11390625,27337500
%N A131892 a(n) is the number of shapes of balanced trees with constant branching factor 6 and n nodes. The node is balanced if the size, measured in nodes, of each pair of its children differ by at most one node.
%H A131892 Alois P. Heinz, <a href="/A131892/b131892.txt">Table of n, a(n) for n = 0..259</a>
%H A131892 Jeffrey Barnett, <a href="http://notatt.com/btree-shapes.pdf">Counting Balanced Tree Shapes</a>
%F A131892 a(0) = a(1) = 1; a(6n+1+m) = (6 choose m) * a(n+1)^m * a(n)^(6-m), where n >= 0 and 0 <= m <= 6.
%p A131892 a:= proc(n) option remember; local m, r; if n<2 then 1 else
%p A131892       r:= iquo(n-1, 6, 'm'); binomial(6, m) *a(r+1)^m *a(r)^(6-m) fi
%p A131892     end:
%p A131892 seq(a(n), n=0..50);  # _Alois P. Heinz_, Apr 10 2013
%t A131892 a[n_, k_] := a[n, k] = Module[{m, r}, If[n < 2 || k == 1, 1, If[k == 0, 0, {r, m} = QuotientRemainder[n - 1, k]; Binomial[k, m]*a[r + 1, k]^m*a[r, k]^(k - m)]]];
%t A131892 a[n_] := a[n, 6];
%t A131892 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jun 04 2018, after _Alois P. Heinz_ *)
%Y A131892 Cf. A110316, A131889, A131890, A131891, A131893.
%Y A131892 Column k=6 of A221857. - _Alois P. Heinz_, Apr 17 2013
%K A131892 easy,nonn
%O A131892 0,3
%A A131892 _Jeffrey Barnett_, Jul 24 2007