A301365 Regular triangle where T(n,k) is the number of strict trees of weight n with k leaves.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 2, 1, 0, 1, 2, 4, 4, 1, 0, 1, 3, 7, 9, 7, 1, 0, 1, 3, 10, 19, 20, 11, 1, 0, 1, 4, 15, 35, 51, 43, 16, 1, 0, 1, 4, 18, 55, 104, 123, 84, 22, 1, 0, 1, 5, 25, 84, 196, 298, 284, 153, 29, 1, 0, 1, 5, 30, 120, 331, 624, 783, 614, 260, 37

Offset

1,12

Comments

A strict tree of weight n > 0 is either a single node of weight n, or a finite sequence of two or more strict trees with strictly decreasing weights summing to n.

Links

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

Example

Triangle begins:
  1
  1   0
  1   1   0
  1   1   1   0
  1   2   2   1   0
  1   2   4   4   1   0
  1   3   7   9   7   1   0
  1   3  10  19  20  11   1   0
  1   4  15  35  51  43  16   1   0
The T(7,3) = 7 strict trees: ((51)1), ((42)1), ((41)2), ((32)2), (4(21)), ((31)3), (421).

Mathematica

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

Prog

(PARI) A(n)={my(v=vector(n)); for(n=1, n, v[n] = y + polcoef(prod(k=1, n-1, 1 + v[k]*x^k + O(x*x^n)), n)); vector(n, k, Vecrev(v[k]/y, k))}
my(T=A(10)); for(n=1, #T, print(T[n])) \\ Andrew Howroyd, Aug 26 2018

Crossrefs

Row sums are A273873.

Cf. A004111, A008284, A032305, A055277, A063834, A281145, A289501, A294018, A294079, A300352, A300442, A300443, A301342, A301364-A301368.

Sequence in context: A309416 A251418 A172101 * A109246 A037890 A037898

Adjacent sequences: A301362 A301363 A301364 * A301366 A301367 A301368

Keyword

nonn,tabl

Author

Gus Wiseman, Mar 19 2018

Status

approved