how decomposition this formula

Question

how decomposition this formula $( x+y)^{2^n}$

$$n=1$$ $$(x+y)^2=x^2+2xy+y^2$$ $$n=2$$ $$(x+y)^4=x^4+4x^3y+6x^2y^2+4xy^3+y^4$$

How to complete to find general formula? Thanks in advance

Answer 1

From the Binomial Formula, $$(x+y)^n=x^n+\binom n1x^{n-1}y+\binom n2x^{n-2}y^2+\cdots+\binom nny^n$$ so replacing $n$ by $2^n$ we have $$(x+y)^{2^n}=\sum_{k=0}^{2^n}\binom{2^n}kx^{2^n-k}y^k$$