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A301366
Regular triangle where T(n,k) is the number of same-trees of weight n with k leaves.
1
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.
Sequence in context: A368842 A089069 A143535 * A250100 A339627 A016270
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Mar 19 2018
STATUS
approved