total number of valence electrons of p block ions

sum_(k=0)^({0, 8, 8, 8, 0, 8, 8, 8, 18, 2, 18, 8, 8, 8, 18, 2, 18, 2, 18, 8}) x^k = {1, x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, 1, x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, x^2 + x + 1, x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, x^2 + x + 1, x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, x^2 + x + 1, x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1}