(1 + x + x^3)^7 = [1 + (x + x^3)]^7
= 1 + 7(x + x^3) + 21(x + x^3)^2 + 35((x + x^3)^3 + 35(x + x^3)^4 + 21(x + x^3)^5 + 7(x + x^3)^6 + (x + x^3)^7
= 1 + 7x + 7x^3 + 21(x^2 + 2x^4 + x^6) + 35(x^3 + 3x^5 + 3x^7 + x^9) + 35(x^4 + 4x^6 + 6x^8 + 4x^10 + x^12) + 21(x^5 + 5x^7 + 10x^9 + 10x^11 + 5x^13 + x^15) + 7(x^6 + 6x^8 + 15x^10 + 20x^12 + 15x^14 + 6x^16 + x^18) + (x^7 + 7x^9 + 21x^11 + 35x^13 + 35x^15 + 21x^17 + 7x^19 + x^21)
= 1+ 7x + 21x^2 + 42x^3 + 77x^4 + 126x^5 + 168x^6 + 211x^7 + 252x^8 + 252x^9 + 245x^10 + 231x^11 + 175x^12 + 140x^13 + 105x14 + 56x^15 + 42x^16 + 21x^17 + 7x^18 + 7x^19 + x^21.