The coefficient of x^k in the expansion of 1-2 x-x^2 e^ -x is

Step-by-step solution

aligned & Now, 1-2 x-x^2 e^ -x & = (1-2 x-x^2 ) (e^x ) & = (1-2 x-x^2 ) (1+x+ x^2 2 ! + x^3 3 ! + . & .+ x^k k ! + ) & = (1+x+ x^2 2 ! + + x^k k ! + )-2 (x+x^2+ x^3 2 ! . & .+ + x^k (k-1) ! + x^ k+1 k ! + ) & - (x^2+x^3+ x^4 2 ! + . . & . x^k (k-2) ! + x^ k+1 (k-1) ! + x^ k+2 k ! + ) & aligned aligned & Coefficient of x^k in ( 1-2 x-x^2 e^ -x ) &= 1 k ! - 2 (k-1) ! - 1 (k-2) ! &= 1 k ! - 2 k k(k-1) ! - k(k-1) k(k-1)(k-2) ! &= 1 k ! - 2 k k ! - k(k-1) k ! &= 1-k-k^2 k ! aligned