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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.

%I A209666 #16 Nov 17 2018 15:25:41
%S A209666 1,2,7,3,18,55,5,50,216,631,7,118,729,2780,8001,11,301,2621,12954,
%T A209666 45865,130453,15,684,8535,55196,241870,820554,2323483,22,1621,28689,
%U A209666 241634,1307055,5280204,17353028,48916087,30,3620,91749,1012196,6783210,32711022,124991685,401709720,1129559068
%N 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.
%H A209666 Alois P. Heinz, <a href="/A209666/b209666.txt">Rows n = 1..141, flattened</a>
%H A209666 Wikipedia, <a href="http://en.wikipedia.org/wiki/Symmetric_polynomials">Symmetric Polynomials</a>
%e A209666 Table starts as:
%e A209666   1;
%e A209666   2,   7;
%e A209666   3,  18, 55;
%e A209666   5,  50, 216,  631;
%e A209666   7, 118, 729, 2780, 8001;
%p A209666 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A209666       add(b(n-i*j, i-1, k)*binomial(i+k-1, k-1)^j, j=0..n/i)))
%p A209666     end:
%p A209666 T:= (n, k)-> b(n$2, k):
%p A209666 seq(seq(T(n, k), k=1..n), n=1..10);  # _Alois P. Heinz_, Mar 04 2016
%t A209666 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}]
%Y A209666 Main diagonal is A209668; row sums are A209667.
%K A209666 nonn,tabl
%O A209666 1,2
%A A209666 _Wouter Meeussen_, Mar 11 2012