A007346 Order of group generated by perfect shuffles of 2n cards. (Formerly M1909)

2, 8, 24, 24, 1920, 7680, 322560, 64, 92897280, 3715891200, 40874803200, 194641920, 25505877196800, 1428329123020800, 21424936845312000, 160, 23310331287699456000, 1678343852714360832000, 31888533201572855808000

Offset

1,1

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Steve Butler, Persi Diaconis and R. L. Graham, The mathematics of the flip and horseshoe shuffles, arXiv:1412.8533 [math.CO], 2014.

Steve Butler, Persi Diaconis and R. L. Graham, The mathematics of the flip and horseshoe shuffles, The American Mathematical Monthly 123.6 (2016): 542-556.

P. Diaconis, R. L. Graham and W. M. Kantor, The mathematics of perfect shuffles, Adv. Appl. Math. 4 (2) (1983) 175-196.

Index entries for sequences related to groups

Formula

See Maple program. - N. J. A. Sloane, Jun 20 2016

Maple

f:=proc(n) local k, i, np;
if n=1 then 2
elif (n mod 2) = 1 then n!*2^(n-1)
elif n=6 then 2^9*3*5
elif n=12 then 2^17*3^3*5*11
elif n=2 then 8
elif (n mod 4)=2 then n!*2^n
else
np:=n; k:=1;
for i while (np mod 2) = 0 do
   np:=np/2; k:=k+1; od;
   if (n=2^(k-1)) then k*2^k else n!*2^(n-2); fi;
fi;
end;
[seq(f(n), n=1..64)]; # N. J. A. Sloane, Jun 20 2016

Mathematica

a[1] = 2; a[2] = 8; a[n_] := With[{m = 2^n*n!}, Which[Mod[n, 4] == 2, If[n == 6, m/6, m], Mod[n, 4] == 1, m/2, Mod[n, 4] == 3, m/2, True, If[n == 2^IntegerExponent[n, 2], 2*n*(IntegerExponent[n, 2] + 1), If[n == 12, m/(2*7!), m/4]]]]; Table[a[n], {n, 1, 19}](* Jean-François Alcover, Feb 17 2012, after Franklin T. Adams-Watters *)

Prog

(PARI) A007346(n) = local(M); M=2^n*n!; if(n%4==2, if(n==2, 8, if(n==6, M/6, M)), if(n%4==1, if(n==1, 2, M/2), if(n%4==3, M/2, if(n==2^valuation(n, 2), 2*n*(valuation(n, 2)+1), if(n==12, M/(7!*2), M/4))))) \\ Franklin T. Adams-Watters, Nov 30 2006

Crossrefs

Cf. A002326, A024222, A274299.

Bisections give A002671, A274303.

Sequence in context: A303861 A138387 A354176 * A062247 A284951 A357635

Adjacent sequences: A007343 A007344 A007345 * A007347 A007348 A007349

Keyword

nonn,nice,easy

Author

N. J. A. Sloane, Mira Bernstein

Extensions

Corrected and extended by Franklin T. Adams-Watters, Nov 30 2006

Status

approved