1
$\begingroup$

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?

$\endgroup$
3
  • $\begingroup$ What are the conditions on $k$ ? Without condition, you have not an integer sequence. $\endgroup$ Mar 4, 2013 at 11:51
  • $\begingroup$ To know how you cama across this sequence might help in answering. $\endgroup$
    – user9072
    Mar 4, 2013 at 13:19
  • $\begingroup$ 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 $\endgroup$ Mar 4, 2013 at 16:45

2 Answers 2

5
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.