A131893 a(n) is the number of shapes of balanced trees with constant branching factor 7 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, 7, 21, 35, 35, 21, 7, 1, 49, 1029, 12005, 84035, 352947, 823543, 823543, 17294403, 155649627, 778248135, 2334744405, 4202539929, 4202539929, 1801088541, 21012699645, 105063498225, 291843050625, 486405084375, 486405084375, 270225046875, 64339296875

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..400

Jeffrey Barnett, Counting Balanced Tree Shapes

Formula

a(0) = a(1) = 1; a(7n+1+m) = (7 choose m) * a(n+1)^m * a(n)^(7-m), where n >= 0 and 0 <= m <= 7.

Maple

a:= proc(n) option remember; local m, r; if n<2 then 1 else
      r:= iquo(n-1, 7, 'm'); binomial(7, m) *a(r+1)^m *a(r)^(7-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, 7];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

Crossrefs

Cf. A110316, A131889, A131890, A131891, A131892.

Column k=7 of A221857. - Alois P. Heinz, Apr 17 2013

Sequence in context: A230210 A087111 A173676 * A282248 A282349 A356454

Adjacent sequences: A131890 A131891 A131892 * A131894 A131895 A131896

Keyword

easy,nonn

Author

Jeffrey Barnett, Jul 24 2007

Status

approved