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A131893 a(n) is the number of shapes of balanced trees with constant branching factor 7 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 A131893 #20 Aug 01 2019 20:27:44
%S A131893 1,1,7,21,35,35,21,7,1,49,1029,12005,84035,352947,823543,823543,
%T A131893 17294403,155649627,778248135,2334744405,4202539929,4202539929,
%U A131893 1801088541,21012699645,105063498225,291843050625,486405084375,486405084375,270225046875,64339296875
%N A131893 a(n) is the number of shapes of balanced trees with constant branching factor 7 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 A131893 Alois P. Heinz, <a href="/A131893/b131893.txt">Table of n, a(n) for n = 0..400</a>
%H A131893 Jeffrey Barnett, <a href="http://notatt.com/btree-shapes.pdf">Counting Balanced Tree Shapes</a>
%F A131893 a(0) = a(1) = 1; a(7n+1+m) = (7 choose m) * a(n+1)^m * a(n)^(7-m), where n >= 0 and 0 <= m <= 7.
%p A131893 a:= proc(n) option remember; local m, r; if n<2 then 1 else
%p A131893       r:= iquo(n-1, 7, 'm'); binomial(7, m) *a(r+1)^m *a(r)^(7-m) fi
%p A131893     end:
%p A131893 seq(a(n), n=0..50);  # _Alois P. Heinz_, Apr 10 2013
%t A131893 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 A131893 a[n_] := a[n, 7];
%t A131893 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jun 04 2018, after _Alois P. Heinz_ *)
%Y A131893 Cf. A110316, A131889, A131890, A131891, A131892.
%Y A131893 Column k=7 of A221857. - _Alois P. Heinz_, Apr 17 2013
%K A131893 easy,nonn
%O A131893 0,3
%A A131893 _Jeffrey Barnett_, Jul 24 2007