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

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, 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, 1, 1, 1, 3, 7, 21, 41, 58, 100, 100, 94, 48, 20

Offset

1,9

Comments

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.

Links

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

Example

Triangle begins:
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  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   1
1   1   3   7  21  41  58 100 100  94  48  20
The T(8,4) = 6 same-trees: (4(2(11))), (4((11)2)), ((22)(22)), ((2(11))4), (((11)2)4), (2222).

Mathematica

sametrees[n_]:=Prepend[Join@@Table[Tuples[sametrees/@ptn], {ptn, Select[IntegerPartitions[n], Length[#]>1&&SameQ@@#&]}], n];
Table[Length[Select[sametrees[n], Count[#, _Integer, {-1}]===k&]], {n, 12}, {k, n}]

Prog

(PARI)
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)}
{my(v=A(16)); for(n=1, #v, print(v[n]))} \\ Andrew Howroyd, Aug 20 2018

Crossrefs

Last entries of each row give A006241. Row sums are A281145.

Cf. A003238, A008284, A055277, A063834, A273873, A289501, A294080, A298422, A298426, A299201, A299203, A300442, A300443, A301343, A301364-A301368.

Sequence in context: A368842 A089069 A143535 * A250100 A339627 A016270

Adjacent sequences: A301363 A301364 A301365 * A301367 A301368 A301369

Keyword

nonn,tabl

Author

Gus Wiseman, Mar 19 2018

Status

approved