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A301365 Regular triangle where T(n,k) is the number of strict trees of weight n with k leaves.

%I A301365 #12 Aug 26 2018 02:34:23
%S A301365 1,1,0,1,1,0,1,1,1,0,1,2,2,1,0,1,2,4,4,1,0,1,3,7,9,7,1,0,1,3,10,19,20,
%T A301365 11,1,0,1,4,15,35,51,43,16,1,0,1,4,18,55,104,123,84,22,1,0,1,5,25,84,
%U A301365 196,298,284,153,29,1,0,1,5,30,120,331,624,783,614,260,37
%N A301365 Regular triangle where T(n,k) is the number of strict trees of weight n with k leaves.
%C A301365 A strict tree of weight n > 0 is either a single node of weight n, or a finite sequence of two or more strict trees with strictly decreasing weights summing to n.
%H A301365 Andrew Howroyd, <a href="/A301365/b301365.txt">Table of n, a(n) for n = 1..1275</a>
%e A301365 Triangle begins:
%e A301365   1
%e A301365   1   0
%e A301365   1   1   0
%e A301365   1   1   1   0
%e A301365   1   2   2   1   0
%e A301365   1   2   4   4   1   0
%e A301365   1   3   7   9   7   1   0
%e A301365   1   3  10  19  20  11   1   0
%e A301365   1   4  15  35  51  43  16   1   0
%e A301365 The T(7,3) = 7 strict trees: ((51)1), ((42)1), ((41)2), ((32)2), (4(21)), ((31)3), (421).
%t A301365 strtrees[n_]:=Prepend[Join@@Table[Tuples[strtrees/@ptn],{ptn,Select[IntegerPartitions[n],Length[#]>1&&UnsameQ@@#&]}],n];
%t A301365 Table[Length[Select[strtrees[n],Count[#,_Integer,{-1}]===k&]],{n,12},{k,n}]
%o A301365 (PARI) A(n)={my(v=vector(n)); for(n=1, n, v[n] = y + polcoef(prod(k=1, n-1, 1 + v[k]*x^k + O(x*x^n)), n)); vector(n, k, Vecrev(v[k]/y, k))}
%o A301365 my(T=A(10));for(n=1, #T, print(T[n])) \\ _Andrew Howroyd_, Aug 26 2018
%Y A301365 Row sums are A273873.
%Y A301365 Cf. A004111, A008284, A032305, A055277, A063834, A281145, A289501, A294018, A294079, A300352, A300442, A300443, A301342, A301364-A301368.
%K A301365 nonn,tabl
%O A301365 1,12
%A A301365 _Gus Wiseman_, Mar 19 2018