A320169 - OEIS

%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