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For an integer $n$, the semi-factorial $n!!$ can be defined as $$ n!! = n(n-2)(n-4)\cdots $$ In other words, the semi-factorial of $n$ is the familiar factorial, but with every other term omitted. For example, $6!! = 6 \cdot 4 \cdot 2$.

This might be generalized slightly as follows: For an integer $n$, the "$k$-semi-factorial" of $n$ is the familiar factorial, but with every $k^\text{th}$ term omitted. For example, the "$3$-semi-factorial" of $6$ would be $6 \cdot 5 \cdot 3 \cdot 2$.

Is the concept of "$k$-semi-factorial" already named? If so, how is it generally denoted?

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I would take 6!! to be $6\cdot 4 \cdot 2$, not $5 \cdot 3 \cdot 1$ Mathworld agrees, but Wikipedia thinks it questionable, though it cites this as one possibility. –  Ross Millikan Oct 12 '12 at 3:05
    
@RossMillikan I think you are correct about the traditional semi-factorial. I've updated my question. –  Austin Mohr Oct 12 '12 at 3:18

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