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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$\begingroup$ What are the conditions on $k$ ? Without condition, you have not an integer sequence. $\endgroup$– Samuele GiraudoCommented Mar 4, 2013 at 11:51
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$\begingroup$ To know how you cama across this sequence might help in answering. $\endgroup$– user9072Commented Mar 4, 2013 at 13:19
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$\begingroup$ A general bit of advice for answering questions of this kind: calculate the first several values, and enter them into the box here: oeis.org $\endgroup$– Mark MeckesCommented Mar 4, 2013 at 16:45
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2 Answers
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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 http://oeis.org/A157000 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 http://en.wikipedia.org/wiki/Menage_problem may be helpful in tracking things down. I don't see an explicit name given to these numbers.
answered Mar 4, 2013 at 16:40