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A301366 Regular triangle where T(n,k) is the number of same-trees of weight n with k leaves.

%I A301366 #9 Aug 20 2018 20:51:55
%S A301366 1,1,1,1,0,1,1,1,2,2,1,0,0,0,1,1,1,1,5,3,3,1,0,0,0,0,0,1,1,1,2,6,12,
%T A301366 14,12,6,1,0,1,0,3,0,3,0,2,1,1,0,0,1,7,10,10,5,3,1,0,0,0,0,0,0,0,0,0,
%U A301366 1,1,1,3,7,21,41,58,100,100,94,48,20
%N A301366 Regular triangle where T(n,k) is the number of same-trees of weight n with k leaves.
%C A301366 A same-tree of weight n > 0 is either a single node of weight n, or a finite sequence of two or more same-trees whose weights are all the same and sum to n.
%H A301366 Andrew Howroyd, <a href="/A301366/b301366.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%e A301366 Triangle begins:
%e A301366 1
%e A301366 1   1
%e A301366 1   0   1
%e A301366 1   1   2   2
%e A301366 1   0   0   0   1
%e A301366 1   1   1   5   3   3
%e A301366 1   0   0   0   0   0   1
%e A301366 1   1   2   6  12  14  12   6
%e A301366 1   0   1   0   3   0   3   0   2
%e A301366 1   1   0   0   1   7  10  10   5   3
%e A301366 1   0   0   0   0   0   0   0   0   0   1
%e A301366 1   1   3   7  21  41  58 100 100  94  48  20
%e A301366 The T(8,4) = 6 same-trees: (4(2(11))), (4((11)2)), ((22)(22)), ((2(11))4), (((11)2)4), (2222).
%t A301366 sametrees[n_]:=Prepend[Join@@Table[Tuples[sametrees/@ptn],{ptn,Select[IntegerPartitions[n],Length[#]>1&&SameQ@@#&]}],n];
%t A301366 Table[Length[Select[sametrees[n],Count[#,_Integer,{-1}]===k&]],{n,12},{k,n}]
%o A301366 (PARI)
%o A301366 A(n)={my(v=vector(n)); for(n=1, n, v[n] = x + sumdiv(n, d, v[n/d]^d)); apply(p -> Vecrev(p/x), v)}
%o A301366 {my(v=A(16)); for(n=1, #v, print(v[n]))} \\ _Andrew Howroyd_, Aug 20 2018
%Y A301366 Last entries of each row give A006241. Row sums are A281145.
%Y A301366 Cf. A003238, A008284, A055277, A063834, A273873, A289501, A294080, A298422, A298426, A299201, A299203, A300442, A300443, A301343, A301364-A301368.
%K A301366 nonn,tabl
%O A301366 1,9
%A A301366 _Gus Wiseman_, Mar 19 2018