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%I #25 May 08 2020 17:31:49
%S 1,1,1,3,23,71,757,2559,30671,1383331,5003791,245273927,3362110459,
%T 12517624987,175179377183,9356953451851,509614686432899,
%U 1938763632210843,107752663194272623
%N Swinging Wilson quotients ((p-1)$ +(-1)^floor((p+2)/2))/p, p prime. Here '$' denotes the swinging factorial function (A056040).
%H G. C. Greubel, <a href="/A163210/b163210.txt">Table of n, a(n) for n = 1..470</a>
%H M. E. Bassett, S. Majid, <a href="http://arxiv.org/abs/1603.00426">Finite noncommutative geometries related to F_p[x]</a>, arXiv:1603.00426 [math.QA], 2016.
%H Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H Peter Luschny, <a href="http://www.luschny.de/math/primes/SwingingPrimes.html"> Swinging Primes.</a>
%e The 5th prime is 11, (11-1)$ = 252, the remainder term is (-1)^floor((11+2)/2)=1. So the quotient (252+1)/11 = 23 is the 5th member of the sequence.
%p swing := proc(n) option remember; if n = 0 then 1 elif irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
%p WQ := proc(f,r,n) map(p->(f(p-1)+r(p))/p,select(isprime,[$1..n])) end:
%p A163210 := n -> WQ(swing,p->(-1)^iquo(p+2,2),n);
%t sf[n_] := n!/Quotient[n, 2]!^2; a[n_] := (p = Prime[n]; (sf[p - 1] + (-1)^Floor[(p + 2)/2])/p); Table[a[n], {n, 1, 19}] (* _Jean-François Alcover_, Jun 28 2013 *)
%t a[p_] := (Binomial[p-1, (p-1)/2] - (-1)^((p-1)/2)) / p
%t Join[{1, 1}, a[Prime[Range[3,20]]]] (* _Peter Luschny_, May 13 2017 *)
%o (PARI) a(n, p=prime(n)) = ((p-1)!/((p-1)\2)!^2 - (-1)^(p\2))/p \\ _David A. Corneth_, May 13 2017
%Y Cf. A163213, A002068, A163212, A163209, A007619, A007540.
%K nonn
%O 1,4
%A _Peter Luschny_, Jul 24 2009
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