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.

1, 1, 5, 10, 10, 5, 1, 25, 250, 1250, 3125, 3125, 31250, 125000, 250000, 250000, 100000, 500000, 1000000, 1000000, 500000, 100000, 250000, 250000, 125000, 31250, 3125, 3125, 1250, 250, 25, 1, 125, 6250, 156250, 1953125, 9765625, 488281250, 9765625000

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..781

Jeffrey Barnett, Counting Balanced Tree Shapes

Formula

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.

Maple

a:= proc(n) option remember; local m, r; if n<2 then 1 else
      r:= iquo(n-1, 5, 'm'); binomial(5, m) *a(r+1)^m *a(r)^(5-m) fi
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 10 2013

Mathematica

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)]]];
a[n_] := a[n, 5];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

Crossrefs

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

Column k=5 of A221857. - Alois P. Heinz, Apr 17 2013

Sequence in context: A277950 A087109 A063261 * A062986 A291380 A280718

Adjacent sequences: A131888 A131889 A131890 * A131892 A131893 A131894

Keyword

easy,nonn

Author

Jeffrey Barnett, Jul 24 2007

Status

approved