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To evaluate $ \sum_{\substack{\text{even }k\\ 2\leq k\leq m-2}} \binom{m-2}{k-1}(k-1) $, where $m$ is an even integer.

Running the following in Mathematica 10.3.0 (October 9, 2015)

Simplify[Sum[Binomial[m - 2, k - 1] (k - 1), {k, 2, m-2, 2}], m/2 \[Element] Integers && m > 10]

gives

0

which is obviously wrong. It may not have a closed form but it shouldn't return 0.

Bug introduced in 9.0 or earlier and persisting through 11.0.1 or later

To evaluate $ \sum_{\substack{\text{even }k\\ 2\leq k\leq m-2}} \binom{m-2}{k-1}(k-1) $, where $m$ is an even integer.

Running the following in Mathematica 10.3.0 (October 9, 2015)

Simplify[Sum[Binomial[m - 2, k - 1] (k - 1), {k, 2, m-2, 2}], m/2 \[Element] Integers && m > 10]

gives

0

which is obviously wrong. It may not have a closed form but it shouldn't return 0.

To evaluate $ \sum_{\substack{\text{even }k\\ 2\leq k\leq m}} \binom{m-2}{k-1}(k-1) $, where $m$ is an even integer.

Running the following in Mathematica 10.3.0 (October 9, 2015)

Simplify[Sum[Binomial[m - 2, k - 1] (k - 1), {k, 2, m, 2}], m/2 \[Element] Integers && m > 10]

gives

0

which is obviously wrong. It may not have a closed form but it shouldn't return 0.

To evaluate $ \sum_{\substack{\text{even }k\\ 2\leq k\leq m-2}} \binom{m-2}{k-1}(k-1) $, where $m$ is an even integer.

Running the following in Mathematica 10.3.0 (October 9, 2015)

Simplify[Sum[Binomial[m - 2, k - 1] (k - 1), {k, 2, m-2, 2}], m/2 \[Element] Integers && m > 10]

gives

0

which is obviously wrong. It may not have a closed form but it shouldn't return 0.