Thank you Jose. I am now able to include the cycle index for SymmetricGroup[1] in expressing certain integer sequences as a sum of symmetric groups. The calculation below is an example for finding the (N-3)/2 nd term for N = 41 in OEIS A000930
In[378]:= A
Out[378]= {19, 16, 13, 10, 7, 4, 1}
In[384]:= Table[
CycleIndexPolynomial[
SymmetricGroup[A[[k]]], {k, k, k, k, k, k, k, k, k, k, k, k, k, k,
k, k, k, k}, A[[k]]], {k, 1, imax + 1}]
Out[384]= {1, 17, 105, 286, 330, 126, 7}
In[385]:= Total[%]
Out[385]= 872