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J, 18 bytes */@,@(1+%)1++/&i./ Try it online! Takes z on its left and x y on the right. Uses modified formula by Peter Taylor: $$ \begin{align} \prod_{i=1}^x \prod_{j=1}^y \frac{i+j+z-1}{i+j-1} &= \prod_{i=1}^x \prod_{j=1}^y \left(1 + \frac{z}{i+j-1}\right) \\ &= \prod_{i=0}^{x-1} \prod_{j=0}^{y-1} \left(1 + \frac{z}{i+j+1}\right) \end{align}...


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x86 machine code, 155 bytes Hexdump: 60 8b fa 51 b0 20 f3 aa 59 51 b0 5f 03 c9 f3 aa 5e b0 0a aa 57 5f 57 b7 2f 8d 14 36 8b ce 2b ca 75 03 80 f7 73 7d 02 f7 d1 51 b0 20 f3 aa 59 8a c7 aa 8b ee 53 6b d9 fe 8d 5c b3 ff 53 6b de fc 8b 5c 1f fe 8b 47 fe 80 fc 5f 74 23 8a c4 34 73 3c 2f 75 0a 80 ff 5f 74 3e 80 ff 5c 74 37 c1 eb 08 66 81 fb 2f 5c 74 2f 0f c7 f3 ...


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