A316624 Number of balanced p-trees with n leaves.

1, 1, 1, 2, 2, 4, 4, 8, 9, 16, 20, 40, 47, 83, 111, 201, 259, 454, 603, 1049, 1432, 2407, 3390, 6006, 8222, 13904, 20304, 34828, 50291, 85817, 126013, 217653, 317894, 535103, 798184, 1367585, 2008125, 3360067, 5048274, 8499942, 12623978, 21023718, 31552560, 52575257

Offset

1,4

Comments

A p-tree of weight n is either a single node (if n = 1) or a finite sequence of p-trees whose weights are weakly decreasing and sum to n.

A tree is balanced if all leaves have the same height.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..500

Example

The a(1) = 1 through a(7) = 4 balanced p-trees:
  o  (oo)  (ooo)  (oooo)      (ooooo)      (oooooo)        (ooooooo)
                  ((oo)(oo))  ((ooo)(oo))  ((ooo)(ooo))    ((oooo)(ooo))
                                           ((oooo)(oo))    ((ooooo)(oo))
                                           ((oo)(oo)(oo))  ((ooo)(oo)(oo))

Mathematica

ptrs[n_]:=If[n==1, {"o"}, Join@@Table[Tuples[ptrs/@p], {p, Rest[IntegerPartitions[n]]}]];
Table[Length[ptrs[n]], {n, 12}]
Table[Length[Select[ptrs[n], SameQ@@Length/@Position[#, "o"]&]], {n, 12}]

Prog

(PARI) seq(n)={my(p=x + O(x*x^n), q=0); while(p, q+=p; p = 1/prod(k=1, n, 1 - polcoef(p, k)*x^k + O(x*x^n)) - 1 - p); Vec(q)} \\ Andrew Howroyd, Oct 26 2018

Crossrefs

Cf. A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A196545, A289501, A319312.

Sequence in context: A306692 A356236 A120803 * A318770 A284613 A000011

Adjacent sequences: A316621 A316622 A316623 * A316625 A316626 A316627

Keyword

nonn

Author

Gus Wiseman, Oct 07 2018

Extensions

Terms a(17) and beyond from Andrew Howroyd, Oct 26 2018

Status

approved