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A131891 a(n) is the number of shapes of balanced trees with constant branching factor 5 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 A131891 #18 Aug 01 2019 18:26:08
%S A131891 1,1,5,10,10,5,1,25,250,1250,3125,3125,31250,125000,250000,250000,
%T A131891 100000,500000,1000000,1000000,500000,100000,250000,250000,125000,
%U A131891 31250,3125,3125,1250,250,25,1,125,6250,156250,1953125,9765625,488281250,9765625000
%N A131891 a(n) is the number of shapes of balanced trees with constant branching factor 5 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 A131891 Alois P. Heinz, <a href="/A131891/b131891.txt">Table of n, a(n) for n = 0..781</a>
%H A131891 Jeffrey Barnett, <a href="http://notatt.com/btree-shapes.pdf">Counting Balanced Tree Shapes</a>
%F A131891 a(0) = a(1) = 1; a(5n+1+m) = (5 choose m) * a(n+1)^m * a(n)^(5-m), where n >= 0 and 0 <= m <= 5.
%p A131891 a:= proc(n) option remember; local m, r; if n<2 then 1 else
%p A131891       r:= iquo(n-1, 5, 'm'); binomial(5, m) *a(r+1)^m *a(r)^(5-m) fi
%p A131891     end:
%p A131891 seq(a(n), n=0..50);  # _Alois P. Heinz_, Apr 10 2013
%t A131891 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 A131891 a[n_] := a[n, 5];
%t A131891 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jun 04 2018, after _Alois P. Heinz_ *)
%Y A131891 Cf. A110316, A131889, A131890, A131892, A131893.
%Y A131891 Column k=5 of A221857. - _Alois P. Heinz_, Apr 17 2013
%K A131891 easy,nonn
%O A131891 0,3
%A A131891 _Jeffrey Barnett_, Jul 24 2007