A209666 T(n,k) = count of degree k monomials in the complete homogeneous symmetric polynomials h(mu,k) summed over all partitions mu of n.

1, 2, 7, 3, 18, 55, 5, 50, 216, 631, 7, 118, 729, 2780, 8001, 11, 301, 2621, 12954, 45865, 130453, 15, 684, 8535, 55196, 241870, 820554, 2323483, 22, 1621, 28689, 241634, 1307055, 5280204, 17353028, 48916087, 30, 3620, 91749, 1012196, 6783210, 32711022, 124991685, 401709720, 1129559068

Offset

1,2

Links

Alois P. Heinz, Rows n = 1..141, flattened

Wikipedia, Symmetric Polynomials

Example

Table starts as:
  1;
  2,   7;
  3,  18, 55;
  5,  50, 216,  631;
  7, 118, 729, 2780, 8001;

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*binomial(i+k-1, k-1)^j, j=0..n/i)))
    end:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Mar 04 2016

Mathematica

h[n_, v_] := Tr@ Apply[Times, Table[Subscript[x, j], {j, v}]^# & /@ Compositions[n, v], {1}]; h[par_?PartitionQ, v_] := Times @@ (h[#, v] & /@ par); Table[Tr[(h[#, k] & /@ Partitions[l]) /. Subscript[x, _] -> 1], {l, 10}, {k, l}]

Crossrefs

Main diagonal is A209668; row sums are A209667.

Sequence in context: A138751 A112303 A336854 * A089124 A210662 A229610

Adjacent sequences: A209663 A209664 A209665 * A209667 A209668 A209669

Keyword

nonn,tabl

Author

Wouter Meeussen, Mar 11 2012

Status

approved