A209667 a(n) = count of monomials, of degrees k=0 to n, in the complete homogeneous symmetric polynomials h(mu,k) summed over all partitions mu of n.

1, 1, 9, 76, 902, 11635, 192205, 3450337, 73128340, 1696862300, 44414258862, 1264163699189, 39640715859359, 1340191402045395, 49097854149726795, 1924982506686743639, 80831323253459088871, 3607487926962810556542, 170964537623741430399076

Offset: 0

Author

Wouter Meeussen, Mar 11 2012

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..250
• Wikipedia, Symmetric Polynomials

Crossrefs

Cf. A209664, A209665, A209666, A209667, A209668, A209669, A209670, A209671, A209672, A209673.

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:
a:= n-> add(b(n$2, k), k=0..n):
seq(a(n), n=0..20);  # 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); Tr/@ Table[Tr[(h[#, k] & /@ Partitions[l]) /. Subscript[x, _] -> 1], {l, 10}, {k, l}]

Formula

Row sums of table A209666.

Extensions

a(0), a(11)-a(18) from Alois P. Heinz, Mar 04 2016