A301368 Regular triangle where T(n,k) is the number of binary enriched p-trees of weight n with k leaves.

1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 2, 4, 5, 3, 1, 3, 7, 12, 12, 6, 1, 3, 9, 19, 28, 25, 11, 1, 4, 14, 36, 65, 81, 63, 24, 1, 4, 16, 48, 107, 172, 193, 136, 47, 1, 5, 22, 75, 192, 369, 522, 522, 331, 103, 1, 5, 25, 96, 284, 643, 1108, 1420, 1292, 750, 214, 1, 6

Offset

1,8

Comments

A binary enriched p-tree of weight n is either a single node of weight n, or an ordered pair of binary enriched p-trees with weakly decreasing weights summing to n.

Links

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

Example

Triangle begins:
  1
  1   1
  1   1   1
  1   2   3   2
  1   2   4   5   3
  1   3   7  12  12   6
  1   3   9  19  28  25  11
  1   4  14  36  65  81  63  24
  1   4  16  48 107 172 193 136  47
  1   5  22  75 192 369 522 522 331 103
  ...
The T(6,3) = 7 binary enriched p-trees: ((41)1), ((32)1), (4(11)), ((31)2), ((22)2), (3(21)), ((21)3).

Mathematica

bintrees[n_]:=Prepend[Join@@Table[Tuples[bintrees/@ptn], {ptn, Select[IntegerPartitions[n], Length[#]===2&]}], n];
Table[Length[Select[bintrees[n], Count[#, _Integer, {-1}]===k&]], {n, 13}, {k, n}]

Prog

(PARI) A(n)={my(v=vector(n)); for(n=1, n, v[n] = y + sum(k=1, n\2, v[k]*v[n-k])); apply(p->Vecrev(p/y), v)}
{ my(T=A(10)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Aug 26 2018

Crossrefs

Last entries of each row give A000992. Row sums are A300443.

Cf. A001190, A008284, A055277, A063834, A196545, A273873, A289501, A292050, A298422, A298426, A300354, A300439, A300442, A301344, A301364-A301367.

Sequence in context: A245436 A285581 A222173 * A198242 A049063 A120894

Adjacent sequences: A301365 A301366 A301367 * A301369 A301370 A301371

Keyword

nonn,tabl

Author

Gus Wiseman, Mar 19 2018

Status

approved