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A301368 Regular triangle where T(n,k) is the number of binary enriched p-trees of weight n with k leaves.

%I A301368 #10 Aug 26 2018 16:35:36
%S A301368 1,1,1,1,1,1,1,2,3,2,1,2,4,5,3,1,3,7,12,12,6,1,3,9,19,28,25,11,1,4,14,
%T A301368 36,65,81,63,24,1,4,16,48,107,172,193,136,47,1,5,22,75,192,369,522,
%U A301368 522,331,103,1,5,25,96,284,643,1108,1420,1292,750,214,1,6
%N A301368 Regular triangle where T(n,k) is the number of binary enriched p-trees of weight n with k leaves.
%C A301368 A binary enriched p-tree of weight n is either a single node of weight n, or an ordered pair of binary enriched p-trees with weakly decreasing weights summing to n.
%H A301368 Andrew Howroyd, <a href="/A301368/b301368.txt">Table of n, a(n) for n = 1..1275</a>
%e A301368 Triangle begins:
%e A301368   1
%e A301368   1   1
%e A301368   1   1   1
%e A301368   1   2   3   2
%e A301368   1   2   4   5   3
%e A301368   1   3   7  12  12   6
%e A301368   1   3   9  19  28  25  11
%e A301368   1   4  14  36  65  81  63  24
%e A301368   1   4  16  48 107 172 193 136  47
%e A301368   1   5  22  75 192 369 522 522 331 103
%e A301368   ...
%e A301368 The T(6,3) = 7 binary enriched p-trees: ((41)1), ((32)1), (4(11)), ((31)2), ((22)2), (3(21)), ((21)3).
%t A301368 bintrees[n_]:=Prepend[Join@@Table[Tuples[bintrees/@ptn],{ptn,Select[IntegerPartitions[n],Length[#]===2&]}],n];
%t A301368 Table[Length[Select[bintrees[n],Count[#,_Integer,{-1}]===k&]],{n,13},{k,n}]
%o A301368 (PARI) A(n)={my(v=vector(n)); for(n=1, n, v[n] = y + sum(k=1, n\2, v[k]*v[n-k])); apply(p->Vecrev(p/y), v)}
%o A301368 { my(T=A(10)); for(n=1, #T, print(T[n])) } \\ _Andrew Howroyd_, Aug 26 2018
%Y A301368 Last entries of each row give A000992. Row sums are A300443.
%Y A301368 Cf. A001190, A008284, A055277, A063834, A196545, A273873, A289501, A292050, A298422, A298426, A300354, A300439, A300442, A301344, A301364-A301367.
%K A301368 nonn,tabl
%O A301368 1,8
%A A301368 _Gus Wiseman_, Mar 19 2018