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 A163210 Swinging Wilson quotients ((p-1)\$ +(-1)^floor((p+2)/2))/p, p prime. Here '\$' denotes the swinging factorial function (A056040). 7
 1, 1, 1, 3, 23, 71, 757, 2559, 30671, 1383331, 5003791, 245273927, 3362110459, 12517624987, 175179377183, 9356953451851, 509614686432899, 1938763632210843, 107752663194272623 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 REFERENCES Peter Luschny, "Divide, swing and conquer the factorial and the lcm{1,2,...,n}", preprint, April 2008. LINKS G. C. Greubel, Table of n, a(n) for n = 1..470 M. E. Bassett, S. Majid, Finite noncommutative geometries related to F_p[x], arXiv:1603.00426 [math.QA], 2016. Peter Luschny, Swinging Primes. EXAMPLE 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. MAPLE 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: WQ := proc(f, r, n) map(p->(f(p-1)+r(p))/p, select(isprime, [\$1..n])) end: A163210 := n -> WQ(swing, p->(-1)^iquo(p+2, 2), n); MATHEMATICA 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 *) a[p_] := (Binomial[p-1, (p-1)/2] - (-1)^((p-1)/2)) / p Join[{1, 1}, a[Prime[Range[3, 20]]]] (* Peter Luschny, May 13 2017 *) PROG (PARI) a(n, p=prime(n)) = ((p-1)!/((p-1)\2)!^2 - (-1)^(p\2))/p \\ David A. Corneth, May 13 2017 CROSSREFS Cf. A163213, A002068, A163212, A163209, A007619, A007540. Sequence in context: A121984 A107177 A096207 * A163211 A126335 A256329 Adjacent sequences:  A163207 A163208 A163209 * A163211 A163212 A163213 KEYWORD nonn AUTHOR Peter Luschny, Jul 24 2009 STATUS approved

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Last modified December 13 18:07 EST 2018. Contains 318086 sequences. (Running on oeis4.)