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%I #10 Aug 20 2018 20:52:10
%S 1,1,1,1,0,1,1,1,1,2,1,0,0,0,1,1,1,1,2,1,3,1,0,0,0,0,0,1,1,1,1,3,4,4,
%T 3,5,1,0,1,0,1,0,1,0,2,1,1,0,0,1,2,1,1,1,3,1,0,0,0,0,0,0,0,0,0,1,1,1,
%U 2,4,5,10,11,14,12,14,7,13,1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,1
%N Regular triangle where T(n,k) is the number of orderless same-trees of weight n with k leaves.
%C An orderless same-tree of weight n > 0 is either a single node of weight n, or a finite multiset of two or more orderless same-trees whose weights are all the same and sum to n.
%H Andrew Howroyd, <a href="/A301367/b301367.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%e Triangle begins:
%e 1
%e 1 1
%e 1 0 1
%e 1 1 1 2
%e 1 0 0 0 1
%e 1 1 1 2 1 3
%e 1 0 0 0 0 0 1
%e 1 1 1 3 4 4 3 5
%e 1 0 1 0 1 0 1 0 2
%e 1 1 0 0 1 2 1 1 1 3
%e 1 0 0 0 0 0 0 0 0 0 1
%e 1 1 2 4 5 10 11 14 12 14 7 13
%e 1 0 0 0 0 0 0 0 0 0 0 0 1
%e 1 1 0 0 0 0 1 2 1 1 1 1 1 3
%e The T(8,5) = 4 orderless same-trees: (4((11)(11))), (4(1111)), ((22)(2(11))), (222(11)).
%t olstrees[n_]:=Prepend[Join@@Table[Select[Tuples[olstrees/@ptn],OrderedQ],{ptn,Select[IntegerPartitions[n],Length[#]>1&&SameQ@@#&]}],n];
%t Table[Length[Select[olstrees[n],Count[#,_Integer,{-1}]===k&]],{n,14},{k,n}]
%o (PARI)
%o S(g, k)={polcoef(exp(sum(i=1, k, x^i*subst(g, y, y^i)/i) + O(x*x^k)), k)}
%o A(n)={my(v=vector(n)); for(n=1, n, v[n] = y + sumdiv(n, d, S(v[n/d], d))); apply(p -> Vecrev(p/y), v)}
%o { my(v=A(16)); for(n=1, #v, print(v[n])) } \\ _Andrew Howroyd_, Aug 20 2018
%Y Last entries of each row give A289079. Row sums are A289078.
%Y Cf. A003238, A006241, A008284, A055277, A063834, A273873, A281145, A289501, A294080, A298422, A298426, A299201, A299203, A301343, A301364-A301368.
%K nonn,tabl
%O 1,10
%A _Gus Wiseman_, Mar 19 2018
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