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Do the numbers of the form $A_k(n)=(-1)^{k+1}\frac{n}{k}\binom{n-k-1}{k-1}$ have a particular name?

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What are the conditions on $k$ ? Without condition, you have not an integer sequence. –  Samuele Giraudo Mar 4 '13 at 11:51
To know how you cama across this sequence might help in answering. –  quid Mar 4 '13 at 13:19
A general bit of advice for answering questions of this kind: calculate the first several values, and enter them into the box here: –  Mark Meckes Mar 4 '13 at 16:45

2 Answers 2

These numbers are sequence A029635 in the OEIS, where they are called the $(1,2)$-Pascal triangle (or Lucas triangle). More precisely, the numbers given there are $T(n,k) = \frac{n+k}{n}\binom nk$, so $A_k(n) = (-1)^{k+1}T(n-k,k)$. They are all integers.

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Setting the minus signs aside, the OP's formula for $A_k(n)$ appears as is in the OEIS entry which gives a reference to page 199 of Riordan's Introduction to Combinatorial Analysis, which refers to a 1943 paper by Kaplansky, "Solution of the 'Probléme des Ménages'," in the Bulletin of the AMS (vol. 49, pp. 784-785). The Wikipedia entry may be helpful in tracking things down. I don't see an explicit name given to these numbers.

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