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A131890 a(n) is the number of shapes of balanced trees with constant branching factor 4 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 A131890 #20 Aug 01 2019 18:19:53
%S A131890 1,1,4,6,4,1,16,96,256,256,1536,3456,3456,1296,3456,3456,1536,256,256,
%T A131890 96,16,1,64,1536,16384,65536,1572864,14155776,56623104,84934656,
%U A131890 905969664,3623878656,6442450944,4294967296,17179869184,25769803776,17179869184,4294967296
%N A131890 a(n) is the number of shapes of balanced trees with constant branching factor 4 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 A131890 Alois P. Heinz, <a href="/A131890/b131890.txt">Table of n, a(n) for n = 0..1365</a>
%H A131890 Jeffrey Barnett, <a href="http://notatt.com/btree-shapes.pdf">Counting Balanced Tree Shapes</a>
%F A131890 a(0) = a(1) = 1; a(4n+1+m) = (4 choose m) * a(n+1)^m * a(n)^(4-m), where n >= 0 and 0 <= m <= 4.
%p A131890 a:= proc(n) option remember; local m, r; if n<2 then 1 else
%p A131890       r:= iquo(n-1, 4, 'm'); binomial(4, m) *a(r+1)^m *a(r)^(4-m) fi
%p A131890     end:
%p A131890 seq(a(n), n=0..50);  # _Alois P. Heinz_, Apr 10 2013
%t A131890 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 A131890 a[n_] := a[n, 4];
%t A131890 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jun 04 2018, after _Alois P. Heinz_ *)
%Y A131890 Cf. A110316, A131889, A131891, A131892, A131893.
%Y A131890 Column k=4 of A221857. - _Alois P. Heinz_, Apr 17 2013
%K A131890 easy,nonn
%O A131890 0,3
%A A131890 _Jeffrey Barnett_, Jul 24 2007