A301367 Regular triangle where T(n,k) is the number of orderless same-trees of weight n with k leaves.

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, 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, 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

Offset: 1

Author

Gus Wiseman, Mar 19 2018

Keywords

• nonn
• tabl

Comments

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.

Examples

			Triangle begins:
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   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   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   2   1   1   1   1   1   3
The T(8,5) = 4 orderless same-trees: (4((11)(11))), (4(1111)), ((22)(2(11))), (222(11)).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Crossrefs

Last entries of each row give A289079. Row sums are A289078.

Cf. A003238, A006241, A008284, A055277, A063834, A273873, A281145, A289501, A294080, A298422, A298426, A299201, A299203, A301343, A301364-A301368.

Programs

Mathematica

olstrees[n_]:=Prepend[Join@@Table[Select[Tuples[olstrees/@ptn],OrderedQ],{ptn,Select[IntegerPartitions[n],Length[#]>1&&SameQ@@#&]}],n];
Table[Length[Select[olstrees[n],Count[#,_Integer,{-1}]===k&]],{n,14},{k,n}]

PARI

S(g, k)={polcoef(exp(sum(i=1, k, x^i*subst(g, y, y^i)/i) + O(x*x^k)), k)}
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)}
{ my(v=A(16)); for(n=1, #v, print(v[n])) } \\ Andrew Howroyd, Aug 20 2018