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A209665 a(n) = count of monomials, degree k=0 to n, in the power sum symmetric polynomials m(mu,k) summed over all partitions mu of n.

%I A209665 #13 Nov 24 2016 17:56:59
%S A209665 1,1,8,56,524,5979,85539,1460752,29112516,661843866,16890042828,
%T A209665 477756925128,14830113520286,501073056287725,18303233207719437,
%U A209665 718663995114727640,30181996254384621880,1349979517537576728657,64065538251202398110415,3215056386968174418054634
%N A209665 a(n) = count of monomials, degree k=0 to n, in the power sum symmetric polynomials m(mu,k) summed over all partitions mu of n.
%C A209665 Row sums of A209664.
%H A209665 Alois P. Heinz, <a href="/A209665/b209665.txt">Table of n, a(n) for n = 0..386</a>
%H A209665 Wikipedia, <a href="http://en.wikipedia.org/wiki/Symmetric_polynomials">Symmetric Polynomials</a>
%p A209665 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A209665       b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
%p A209665     end:
%p A209665 a:= n-> add(b(n$2, k), k=0..n):
%p A209665 seq(a(n), n=0..20);  # _Alois P. Heinz_, Nov 24 2016
%t A209665 p[n_Integer, v_] := Sum[Subscript[x, j]^n, {j, v}]; p[par_?PartitionQ, v_] := Times @@ (p[#, v] & /@ par); Tr/@ Table[Tr[(p[#, k] & /@ Partitions[l]) /. Subscript[x, _] -> 1], {l, 11}, {k, l}]
%Y A209665 Cf. A209664.
%K A209665 nonn
%O A209665 0,3
%A A209665 _Wouter Meeussen_, Mar 11 2012
%E A209665 a(0), a(12)-a(19) from _Alois P. Heinz_, Nov 24 2016