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%I #10 Aug 26 2018 16:35:16
%S 1,1,1,1,1,2,1,2,4,5,1,2,6,11,12,1,3,10,26,38,34,1,3,13,39,87,117,92,
%T 1,4,19,69,181,339,406,277,1,4,23,95,303,707,1198,1311,806,1,5,30,143,
%U 514,1430,2970,4525,4522,2500,1,5,35,184,762,2446,6124,11627
%N Regular triangle where T(n,k) is the number of enriched p-trees of weight n with k leaves.
%C An enriched p-tree of weight n > 0 is either a single node of weight n, or a finite sequence of two or more enriched p-trees with weakly decreasing weights summing to n.
%H Andrew Howroyd, <a href="/A301364/b301364.txt">Table of n, a(n) for n = 1..1275</a>
%e Triangle begins:
%e 1
%e 1 1
%e 1 1 2
%e 1 2 4 5
%e 1 2 6 11 12
%e 1 3 10 26 38 34
%e 1 3 13 39 87 117 92
%e 1 4 19 69 181 339 406 277
%e ...
%e The T(5,4) = 11 enriched p-trees: (((21)1)1), ((2(11))1), (((11)2)1), ((211)1), ((21)(11)), (((11)1)2), ((111)2), ((21)11), (2(11)1), ((11)21), (2111).
%t eptrees[n_]:=Prepend[Join@@Table[Tuples[eptrees/@ptn],{ptn,Select[IntegerPartitions[n],Length[#]>1&]}],n];
%t Table[Length[Select[eptrees[n],Count[#,_Integer,{-1}]===k&]],{n,8},{k,n}]
%o (PARI) A(n)={my(v=vector(n)); for(n=1, n, v[n] = y + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x*x^n)), n)); apply(p->Vecrev(p/y), v)}
%o { my(T=A(10)); for(n=1, #T, print(T[n])) } \\ _Andrew Howroyd_, Aug 26 2018
%Y Last entries of each row give A196545. Row sums are A289501.
%Y Cf. A008284, A055277, A063834, A220418, A273866, A273873, A281145, A290261, A299201, A299202, A299203, A300354, A300442, A300443, A301365-A301368.
%K nonn,tabl
%O 1,6
%A _Gus Wiseman_, Mar 19 2018
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