%I #8 Oct 26 2018 00:52:17
%S 1,2,3,6,9,20,31,70,114,243,415,961,1603,3564,6559,14913,26630,60037,
%T 110160,248859,458445,1001190,1882350,4220358,7765303,16822107,
%U 32307240,70081784,133716083,291788153,561823990,1230204229,2396185727,5176454708,10220127290
%N Number of balanced enriched p-trees of weight n.
%C 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.
%C A tree is balanced if all leaves have the same height.
%H Andrew Howroyd, <a href="/A320169/b320169.txt">Table of n, a(n) for n = 1..500</a>
%e The a(1) = 1 through a(6) = 20 balanced enriched p-trees:
%e 1 2 3 4 5 6
%e (11) (21) (22) (32) (33)
%e (111) (31) (41) (42)
%e (211) (221) (51)
%e (1111) (311) (222)
%e ((11)(11)) (2111) (321)
%e (11111) (411)
%e ((21)(11)) (2211)
%e ((111)(11)) (3111)
%e (21111)
%e (111111)
%e ((21)(21))
%e ((22)(11))
%e ((31)(11))
%e ((111)(21))
%e ((21)(111))
%e ((211)(11))
%e ((111)(111))
%e ((1111)(11))
%e ((11)(11)(11))
%t eptrs[n_]:=Prepend[Join@@Table[Tuples[eptrs/@p],{p,Rest[IntegerPartitions[n]]}],n];
%t Table[Length[Select[eptrs[n],SameQ@@Length/@Position[#,_Integer]&]],{n,12}]
%o (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
%Y Cf. A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A196545, A289501, A319312.
%Y Cf. A316624, A320154, A320155, A320160, A320179.
%K nonn
%O 1,2
%A _Gus Wiseman_, Oct 07 2018
%E Terms a(16) and beyond from _Andrew Howroyd_, Oct 26 2018