[seqfan] A combinatorial problem

[Vladimir Shevelev]

Dear SeqFans,
 
I ask anyone to extend a sequence which is connected with the following modification of the menage problem. A well known mathematician N found himself with his wife among the guests, which were
n(>=3) married couples. After seating the ladies on every other chair at a circular table, N was the first offered to choose an arbitrary chair but not side by side with his wife. For which values of n the number of ways of seating of other men ( under the condition that no husband is beside his wife) does not depend on how far N takes his seat from his wife?
 
The first terms of this sequence are 3,4,6.  I proved that the problem reduces to description the values of n>=3 for which, for every r=1,...,n, we have
Sum{k=0,...,n-1}((-1)^k)*(n-k-1)!*B=A000179(n)/(n-2),
where B=Sum{i=0,...,k}C(2r-i-4, i)*C(2n-2r-k+i+2, k-i), i.e., for such an n, B does not depend on r  (here C-binomial coefficients).
In addition, I proved that A000179(n)/(n-2) is integer, if n has the form 2^t+2 ( and I conjecture that here one can write "iff").
 
E.g., if n=3, then, for every r,  if k=0, then B=1; if k=1, then B=2; if k=2, then B=1. Thus
1*2!-2*1!+1*0!=A000179(3)/1=1.
 
Regards,
Vladimir

 Shevelev Vladimir‎