A163590 Odd part of the swinging factorial A056040.

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

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