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

Author

Gus Wiseman, Jan 31 2020

Keywords

• nonn

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.

Examples

			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)
		

Links

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

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.

Programs

Mathematica

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

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

Extensions

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