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

Author

Wouter Meeussen, Mar 11 2012

Keywords

• nonn
• tabl

Examples

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

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• Wikipedia, Symmetric Polynomials

Crossrefs

Main diagonal is A209668; row sums are A209667.

Programs

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}]