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

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