A131892 a(n) is the number of shapes of balanced trees with constant branching factor 6 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, 6, 15, 20, 15, 6, 1, 36, 540, 4320, 19440, 46656, 46656, 699840, 4374000, 14580000, 27337500, 27337500, 11390625, 91125000, 303750000, 540000000, 540000000, 288000000, 64000000, 288000000, 540000000, 540000000, 303750000, 91125000, 11390625, 27337500

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..259

Jeffrey Barnett, Counting Balanced Tree Shapes

Formula

a(0) = a(1) = 1; a(6n+1+m) = (6 choose m) * a(n+1)^m * a(n)^(6-m), where n >= 0 and 0 <= m <= 6.

Maple

a:= proc(n) option remember; local m, r; if n<2 then 1 else
      r:= iquo(n-1, 6, 'm'); binomial(6, m) *a(r+1)^m *a(r)^(6-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, 6];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

Crossrefs

Cf. A110316, A131889, A131890, A131891, A131893.

Column k=6 of A221857. - Alois P. Heinz, Apr 17 2013

Sequence in context: A176849 A087110 A063266 * A291381 A280719 A282173

Adjacent sequences: A131889 A131890 A131891 * A131893 A131894 A131895

Keyword

easy,nonn

Author

Jeffrey Barnett, Jul 24 2007

Status

approved