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

%I A331875 #8 Feb 09 2020 15:18:09
%S A331875 1,1,2,3,6,14,32,79,198,522,1368,3716,9992,27612,75692,212045,589478,
%T A331875 1668630,4690792,13387332,37980664,109098556,311717768,900846484,
%U A331875 2589449032,7515759012,21720369476,63305262126,183726039404,537364221200,1565570459800,4592892152163
%N A331875 Number of enriched identity p-trees of weight n.
%C A331875 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.
%H A331875 Andrew Howroyd, <a href="/A331875/b331875.txt">Table of n, a(n) for n = 1..500</a>
%e A331875 The a(1) = 1 through a(6) = 14 enriched p-trees:
%e A331875   1  2  3     4        5           6
%e A331875         (21)  (31)     (32)        (42)
%e A331875               ((21)1)  (41)        (51)
%e A331875                        ((21)2)     (321)
%e A331875                        ((31)1)     ((21)3)
%e A331875                        (((21)1)1)  ((31)2)
%e A331875                                    ((32)1)
%e A331875                                    (3(21))
%e A331875                                    ((41)1)
%e A331875                                    ((21)21)
%e A331875                                    (((21)1)2)
%e A331875                                    (((21)2)1)
%e A331875                                    (((31)1)1)
%e A331875                                    ((((21)1)1)1)
%t A331875 eptrid[n_]:=Prepend[Select[Join@@Table[Tuples[eptrid/@p],{p,Rest[IntegerPartitions[n]]}],UnsameQ@@#&],n];
%t A331875 Table[Length[eptrid[n]],{n,10}]
%o A331875 (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
%Y A331875 The orderless version is A300660.
%Y A331875 The locally disjoint case is A331684.
%Y A331875 Identity trees are A004111.
%Y A331875 P-trees are A196545.
%Y A331875 Enriched p-trees are A289501.
%Y A331875 Cf. A000669, A141268, A306200, A316471, A331683, A331685, A331686, A331783.
%K A331875 nonn
%O A331875 1,3
%A A331875 _Gus Wiseman_, Jan 31 2020
%E A331875 Terms a(21) and beyond from _Andrew Howroyd_, Feb 09 2020