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Regular triangle where T(n,k) is the number of strict trees of weight n with k leaves.
2

%I #12 Aug 26 2018 02:34:23

%S 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 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 196,298,284,153,29,1,0,1,5,30,120,331,624,783,614,260,37

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

%C 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 Andrew Howroyd, <a href="/A301365/b301365.txt">Table of n, a(n) for n = 1..1275</a>

%e Triangle begins:

%e 1

%e 1 0

%e 1 1 0

%e 1 1 1 0

%e 1 2 2 1 0

%e 1 2 4 4 1 0

%e 1 3 7 9 7 1 0

%e 1 3 10 19 20 11 1 0

%e 1 4 15 35 51 43 16 1 0

%e The T(7,3) = 7 strict trees: ((51)1), ((42)1), ((41)2), ((32)2), (4(21)), ((31)3), (421).

%t strtrees[n_]:=Prepend[Join@@Table[Tuples[strtrees/@ptn],{ptn,Select[IntegerPartitions[n],Length[#]>1&&UnsameQ@@#&]}],n];

%t Table[Length[Select[strtrees[n],Count[#,_Integer,{-1}]===k&]],{n,12},{k,n}]

%o (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 my(T=A(10));for(n=1, #T, print(T[n])) \\ _Andrew Howroyd_, Aug 26 2018

%Y Row sums are A273873.

%Y Cf. A004111, A008284, A032305, A055277, A063834, A281145, A289501, A294018, A294079, A300352, A300442, A300443, A301342, A301364-A301368.

%K nonn,tabl

%O 1,12

%A _Gus Wiseman_, Mar 19 2018