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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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. |
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