Counting Queen's paths on a rectangular chessboard Task

Given two positive integers \$m,n\$, imagine a chessboard of size \$m \times n\$. A chess queen is on the upper-left corner. In how many ways can it reach the lower-right corner, by moving only right, down, or diagonally right-down (possibly moving many steps at once, because it's a queen)?

The resulting 2D sequence is A132439.

Standard code-golf rules apply. The shortest code in bytes wins.

Test cases

m n result
1 1      1
1 2      1
1 3      2
1 4      4
1 5      8
2 2      3
2 3      7
2 4     17
2 5     40
3 3     22
3 4     60
3 5    158
4 4    188
4 5    543
5 5   1712
Haskell, 94 91 64 54 bytes



1!1=1;m!n=sum[m!(n-x)+(m-x)!n+(m-x)!(n-x)|x&lt;-[1..m+n]]


Try it online!

Initially a boring port of the comment on OEIS

-27 from some tricks off dingledooper's answer

-10 from Christian Sievers