A331875 Number of enriched identity p-trees of weight n.

1, 1, 2, 3, 6, 14, 32, 79, 198, 522, 1368, 3716, 9992, 27612, 75692, 212045, 589478, 1668630, 4690792, 13387332, 37980664, 109098556, 311717768, 900846484, 2589449032, 7515759012, 21720369476, 63305262126, 183726039404, 537364221200, 1565570459800, 4592892152163

Offset

1,3

Comments

An enriched identity p-tree of weight n is either the number n itself or a finite sequence of distinct enriched identity p-trees whose weights are weakly decreasing and sum to n.

Links

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

Example

The a(1) = 1 through a(6) = 14 enriched p-trees:
  1  2  3     4        5           6
        (21)  (31)     (32)        (42)
              ((21)1)  (41)        (51)
                       ((21)2)     (321)
                       ((31)1)     ((21)3)
                       (((21)1)1)  ((31)2)
                                   ((32)1)
                                   (3(21))
                                   ((41)1)
                                   ((21)21)
                                   (((21)1)2)
                                   (((21)2)1)
                                   (((31)1)1)
                                   ((((21)1)1)1)

Mathematica

eptrid[n_]:=Prepend[Select[Join@@Table[Tuples[eptrid/@p], {p, Rest[IntegerPartitions[n]]}], UnsameQ@@#&], n];
Table[Length[eptrid[n]], {n, 10}]

Prog

(PARI) seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + polcoef(prod(k=1, n-1, sum(j=0, n\k, j!*binomial(v[k], j)*x^(k*j)) + O(x*x^n)), n)); v} \\ Andrew Howroyd, Feb 09 2020

Crossrefs

The orderless version is A300660.

The locally disjoint case is A331684.

Identity trees are A004111.

P-trees are A196545.

Enriched p-trees are A289501.

Cf. A000669, A141268, A306200, A316471, A331683, A331685, A331686, A331783.

Sequence in context: A307231 A099968 A291401 * A010357 A190166 A238823

Adjacent sequences: A331872 A331873 A331874 * A331876 A331877 A331878

Keyword

nonn

Author

Gus Wiseman, Jan 31 2020

Extensions

Terms a(21) and beyond from Andrew Howroyd, Feb 09 2020

Status

approved