A209667 - OEIS

%I #16 Nov 17 2018 15:25:19

%S 1,1,9,76,902,11635,192205,3450337,73128340,1696862300,44414258862,

%T 1264163699189,39640715859359,1340191402045395,49097854149726795,

%U 1924982506686743639,80831323253459088871,3607487926962810556542,170964537623741430399076

%N 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.

%H Alois P. Heinz, <a href="/A209667/b209667.txt">Table of n, a(n) for n = 0..250</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Symmetric_polynomials">Symmetric Polynomials</a>

%F Row sums of table A209666.

%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p       add(b(n-i*j, i-1, k)*binomial(i+k-1, k-1)^j, j=0..n/i)))

%p     end:

%p a:= n-> add(b(n$2, k), k=0..n):

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Mar 04 2016

%t 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}]

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

%K nonn

%O 0,3

%A _Wouter Meeussen_, Mar 11 2012

%E a(0), a(11)-a(18) from _Alois P. Heinz_, Mar 04 2016