A209666 - OEIS

%I #16 Nov 17 2018 15:25:41

%S 1,2,7,3,18,55,5,50,216,631,7,118,729,2780,8001,11,301,2621,12954,

%T 45865,130453,15,684,8535,55196,241870,820554,2323483,22,1621,28689,

%U 241634,1307055,5280204,17353028,48916087,30,3620,91749,1012196,6783210,32711022,124991685,401709720,1129559068

%N 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 Alois P. Heinz, <a href="/A209666/b209666.txt">Rows n = 1..141, flattened</a>

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

%e Table starts as:

%e   1;

%e   2,   7;

%e   3,  18, 55;

%e   5,  50, 216,  631;

%e   7, 118, 729, 2780, 8001;

%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 T:= (n, k)-> b(n$2, k):

%p seq(seq(T(n, k), k=1..n), n=1..10);  # _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); Table[Tr[(h[#, k] & /@ Partitions[l]) /. Subscript[x, _] -> 1], {l, 10}, {k, l}]

%Y Main diagonal is A209668; row sums are A209667.

%K nonn,tabl

%O 1,2

%A _Wouter Meeussen_, Mar 11 2012