A320169 Number of balanced enriched p-trees of weight n.

1, 2, 3, 6, 9, 20, 31, 70, 114, 243, 415, 961, 1603, 3564, 6559, 14913, 26630, 60037, 110160, 248859, 458445, 1001190, 1882350, 4220358, 7765303, 16822107, 32307240, 70081784, 133716083, 291788153, 561823990, 1230204229, 2396185727, 5176454708, 10220127290

Offset

1,2

Comments

An enriched p-tree of weight n is either the number n itself or a finite sequence of enriched 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(6) = 20 balanced enriched p-trees:
  1  2     3      4           5            6
     (11)  (21)   (22)        (32)         (33)
           (111)  (31)        (41)         (42)
                  (211)       (221)        (51)
                  (1111)      (311)        (222)
                  ((11)(11))  (2111)       (321)
                              (11111)      (411)
                              ((21)(11))   (2211)
                              ((111)(11))  (3111)
                                           (21111)
                                           (111111)
                                           ((21)(21))
                                           ((22)(11))
                                           ((31)(11))
                                           ((111)(21))
                                           ((21)(111))
                                           ((211)(11))
                                           ((111)(111))
                                           ((1111)(11))
                                           ((11)(11)(11))

Mathematica

eptrs[n_]:=Prepend[Join@@Table[Tuples[eptrs/@p], {p, Rest[IntegerPartitions[n]]}], n];
Table[Length[Select[eptrs[n], SameQ@@Length/@Position[#, _Integer]&]], {n, 12}]

Prog

(PARI) seq(n)={my(p=x/(1-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.

Cf. A316624, A320154, A320155, A320160, A320179.

Sequence in context: A320160 A055873 A246565 * A293606 A185376 A321484

Adjacent sequences: A320166 A320167 A320168 * A320170 A320171 A320172

Keyword

nonn

Author

Gus Wiseman, Oct 07 2018

Extensions

Terms a(16) and beyond from Andrew Howroyd, Oct 26 2018

Status

approved