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 A163590 Odd part of the swinging factorial A056040. 4
 1, 1, 1, 3, 3, 15, 5, 35, 35, 315, 63, 693, 231, 3003, 429, 6435, 6435, 109395, 12155, 230945, 46189, 969969, 88179, 2028117, 676039, 16900975, 1300075, 35102025, 5014575, 145422675, 9694845, 300540195 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Let n\$ denote the swinging factorial. a(n) = n\$ / 2^sigma(n) where sigma(n) is the exponent of 2 in the prime-factorization of n\$. sigma(n) can be computed as the number of '1's in the base 2 representation of floor(n/2). 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 = 0..1000 Peter Luschny, Swinging Factorial. FORMULA a(2*n) = A001790(n). a(2*n+1) = A001803(n). EXAMPLE 11\$ = 2772 = 2^2*3^2*7*11. Therefore a(11) = 3^2*7*11 = 2772/4 = 693. 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: sigma := n -> 2^(add(i, i= convert(iquo(n, 2), base, 2))): a := n -> swing(n)/sigma(n); MATHEMATICA sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/ f!]; a[n_] := With[{s = sf[n]}, s/2^IntegerExponent[s, 2]]; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Jul 26 2013 *) PROG (Sage) @CachedFunction def swing(n):     if n == 0: return 1     return swing(n-1)*n if is_odd(n) else 4*swing(n-1)/n A163590 = lambda n: swing(n)/2^A000120(n//2) [A163590(n) for n in (0..31)]  # Peter Luschny, Nov 19 2012 CROSSREFS Cf. A056040, A060632, A001790, A001803. Sequence in context: A165405 A179857 A260078 * A114320 A185138 A285947 Adjacent sequences:  A163587 A163588 A163589 * A163591 A163592 A163593 KEYWORD nonn AUTHOR Peter Luschny, Aug 01 2009 STATUS approved

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Last modified August 20 17:05 EDT 2017. Contains 290836 sequences.