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%I #33 Oct 26 2018 00:51:47
%S 1,1,1,2,2,4,4,8,9,16,20,40,47,83,111,201,259,454,603,1049,1432,2407,
%T 3390,6006,8222,13904,20304,34828,50291,85817,126013,217653,317894,
%U 535103,798184,1367585,2008125,3360067,5048274,8499942,12623978,21023718,31552560,52575257
%N Number of balanced p-trees with n leaves.
%C A p-tree of weight n is either a single node (if n = 1) or a finite sequence of 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="/A316624/b316624.txt">Table of n, a(n) for n = 1..500</a>
%e The a(1) = 1 through a(7) = 4 balanced p-trees:
%e o (oo) (ooo) (oooo) (ooooo) (oooooo) (ooooooo)
%e ((oo)(oo)) ((ooo)(oo)) ((ooo)(ooo)) ((oooo)(ooo))
%e ((oooo)(oo)) ((ooooo)(oo))
%e ((oo)(oo)(oo)) ((ooo)(oo)(oo))
%t ptrs[n_]:=If[n==1,{"o"},Join@@Table[Tuples[ptrs/@p],{p,Rest[IntegerPartitions[n]]}]];
%t Table[Length[ptrs[n]],{n,12}]
%t Table[Length[Select[ptrs[n],SameQ@@Length/@Position[#,"o"]&]],{n,12}]
%o (PARI) seq(n)={my(p=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.
%K nonn
%O 1,4
%A _Gus Wiseman_, Oct 07 2018
%E Terms a(17) and beyond from _Andrew Howroyd_, Oct 26 2018
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