欧拉项目第15题 Lattice paths

Starting in the top left corner of a 2×2 grid, and only being able to move to the right and down, there are exactly 6 routes to the bottom right corner.



How many such routes are there through a 20×20 grid?
从左上开始，只允许往右和往下走，20*20的格子到右下角有多少中走法。

思路如下，将格子看成二维数组，如2*2格子，x[3][3]，从x[0][0] 走到x[3][3]，每次x[i][j]，i加1或者j加1。
容易得出x[0][j]、x[i][0] = 1, x[i][j] = x[i-1][j] + x[i][j-1],  x[i][j] = x[j][i]。
public static void main(String... args) {
Long[][] xx = new Long[21][21];
for(int i=0;i<21;i++){
for(int j=i;j < 21;j++) {
if(i == 0 || j==0) {
xx[i][j] = 1l;
xx[j][i] = xx[i][j];
continue;
} else if(xx[j][i] != null) {
xx[i][j] = xx[j][i];
} else {
xx[i][j] = xx[i-1][j] + xx[i][j-1];
xx[j][i] = xx[i][j];
}
}
}
System.out.println(xx[20][20]);
}
用integer中间会出现溢出，运行错误好几次，差点怀疑自己的思路。。。