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A301367 Regular triangle where T(n,k) is the number of orderless same-trees of weight n with k leaves. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,10

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.

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   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)).

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}]

PROG

(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

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.

Sequence in context: A213731 A299197 A143111 * A179195 A238386 A064662

Adjacent sequences:  A301364 A301365 A301366 * A301368 A301369 A301370

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Mar 19 2018

STATUS

approved

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Last modified October 15 13:38 EDT 2019. Contains 328030 sequences. (Running on oeis4.)