A131891 - OEIS

%I #18 Aug 01 2019 18:26:08

%S 1,1,5,10,10,5,1,25,250,1250,3125,3125,31250,125000,250000,250000,

%T 100000,500000,1000000,1000000,500000,100000,250000,250000,125000,

%U 31250,3125,3125,1250,250,25,1,125,6250,156250,1953125,9765625,488281250,9765625000

%N 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 Alois P. Heinz, <a href="/A131891/b131891.txt">Table of n, a(n) for n = 0..781</a>

%H Jeffrey Barnett, <a href="http://notatt.com/btree-shapes.pdf">Counting Balanced Tree Shapes</a>

%F 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 a:= proc(n) option remember; local m, r; if n<2 then 1 else

%p       r:= iquo(n-1, 5, 'm'); binomial(5, m) *a(r+1)^m *a(r)^(5-m) fi

%p     end:

%p seq(a(n), n=0..50);  # _Alois P. Heinz_, Apr 10 2013

%t 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 a[n_] := a[n, 5];

%t Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jun 04 2018, after _Alois P. Heinz_ *)

%Y Cf. A110316, A131889, A131890, A131892, A131893.

%Y Column k=5 of A221857. - _Alois P. Heinz_, Apr 17 2013

%K easy,nonn

%O 0,3

%A _Jeffrey Barnett_, Jul 24 2007