A301364 Regular triangle where T(n,k) is the number of enriched p-trees of weight n with k leaves.

1, 1, 1, 1, 1, 2, 1, 2, 4, 5, 1, 2, 6, 11, 12, 1, 3, 10, 26, 38, 34, 1, 3, 13, 39, 87, 117, 92, 1, 4, 19, 69, 181, 339, 406, 277, 1, 4, 23, 95, 303, 707, 1198, 1311, 806, 1, 5, 30, 143, 514, 1430, 2970, 4525, 4522, 2500, 1, 5, 35, 184, 762, 2446, 6124, 11627

Offset

1,6

Comments

An enriched p-tree of weight n > 0 is either a single node of weight n, or a finite sequence of two or more 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   2
  1   2   4   5
  1   2   6  11  12
  1   3  10  26  38  34
  1   3  13  39  87 117  92
  1   4  19  69 181 339 406 277
  ...
The T(5,4) = 11 enriched p-trees: (((21)1)1), ((2(11))1), (((11)2)1), ((211)1), ((21)(11)), (((11)1)2), ((111)2), ((21)11), (2(11)1), ((11)21), (2111).

Mathematica

eptrees[n_]:=Prepend[Join@@Table[Tuples[eptrees/@ptn], {ptn, Select[IntegerPartitions[n], Length[#]>1&]}], n];
Table[Length[Select[eptrees[n], Count[#, _Integer, {-1}]===k&]], {n, 8}, {k, n}]

Prog

(PARI) A(n)={my(v=vector(n)); for(n=1, n, v[n] = y + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x*x^n)), n)); 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 A196545. Row sums are A289501.

Cf. A008284, A055277, A063834, A220418, A273866, A273873, A281145, A290261, A299201, A299202, A299203, A300354, A300442, A300443, A301365-A301368.

Sequence in context: A256188 A072727 A292601 * A309503 A057061 A307729

Adjacent sequences: A301361 A301362 A301363 * A301365 A301366 A301367

Keyword

nonn,tabl

Author

Gus Wiseman, Mar 19 2018

Status

approved