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 A163590 Odd part of the swinging factorial A056040. 7
 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, 300540195, 9917826435, 583401555, 20419054425, 2268783825 (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). If n is even then a(n) is the numerator of the reduced ratio (n-1)!!/n!! = A001147(n-1)/A000165(n), and if n is odd then a(n) is the numerator of the reduced ratio n!!/(n-1)!! = A001147(n)/A000165(n-1). The denominators for each ratio should be compared to A060818. Here all ratios are reduced. - Anthony Hernandez, Feb 05 2020 [See the Mathematica program for a more compact form of the formula. Peter Luschny, Mar 01 2020 ] LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011. Peter Luschny, Swinging Factorial. FORMULA a(2*n) = A001790(n). a(2*n+1) = A001803(n). a(n) = a(n-1)*n^((-1)^(n+1))*2^valuation(n, 2) for n > 0. - Peter Luschny, Sep 29 2019 EXAMPLE 11\$ = 2772 = 2^2*3^2*7*11. Therefore a(11) = 3^2*7*11 = 2772/4 = 693. From Anthony Hernandez, Feb 04 2019: (Start) a(7) = numerator((1*3*5*7)/(2*4*6)) = 35; a(8) = numerator((1*3*5*7)/(2*4*6*8)) = 35; a(9) = numerator((1*3*5*7*9)/(2*4*6*8)) = 315; a(10) = numerator((1*3*5*7*9)/(2*4*6*8*10)) = 63. (End) 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 *) r[n_] := (n - Mod[n - 1, 2])!! /(n - 1 + Mod[n - 1, 2])!! ; Table[r[n], {n, 0, 36}] // Numerator (* Peter Luschny, Mar 01 2020 *) PROG (Sage) # uses[A000120] @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 # Alternatively: (Sage) @cached_function def A163590(n): if n == 0: return 1 return A163590(n - 1) * n^((-1)^(n + 1)) * 2^valuation(n, 2) print([A163590(n) for n in (0..31)]) # Peter Luschny, Sep 29 2019 (PARI) A163590(n) = { my(a = vector(n+1)); a[1] = 1; for(n = 1, n, a[n+1] = a[n]*n^((-1)^(n+1))*2^valuation(n, 2)); a } \\ Peter Luschny, Sep 29 2019 CROSSREFS Cf. A056040, A060632, A001790, A001803. Sequence in context: A165405 A179857 A260078 * A344850 A114320 A185138 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 13 13:30 EDT 2024. Contains 375142 sequences. (Running on oeis4.)