in reply to Challenge: Number of unique ways to reach target sum

For symmetry reasons, you get the same result as if you calculate the number of ways you can generate 10 * 101 - 667 = 343 as the sum of 10 numbers in 1 .. 100.

This can speed the calculation to twice three times faster with a minimal change in the program, and eight four times faster if you
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Here's the resulting faster variant of my script: (Update: I've removed the redundant code from this snippet, as noted on Re: Challenge: Number of unique ways to reach target sum and updated the benchmarks accordingly)

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This of course works only in those solutions which calculate the number of possibilities only, not for those that iterate on them all.

Update: blokhead's reply has another interesting optimization.

It uses the fact that if if t is too small or too large, there'll be no solutions. (Too small means t < c * (c + 1) / 2.) I'm not doing that, but at least I check for t being too small when I'm calculating a value. I don't have to check for it being too large explicitly, as that's taken care by the symmetry. This optimization gives a 20% 15% speed improvement. Some other optimizations (not that the big ifelse statement has disappeared completely) give some extra improvement,
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Update:

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like blokhead does makes it even more faster (with 40%). Here's the newer code.
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Update: Improving this a tad bit more, we get

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