A133221 A001147 with each term repeated.

1, 1, 1, 1, 3, 3, 15, 15, 105, 105, 945, 945, 10395, 10395, 135135, 135135, 2027025, 2027025, 34459425, 34459425, 654729075, 654729075, 13749310575, 13749310575, 316234143225, 316234143225, 7905853580625, 7905853580625, 213458046676875, 213458046676875

Offset: 0

Author

N. J. A. Sloane, Oct 13 2007

Keywords

• nonn

Comments

Normally such sequences are excluded from the OEIS, but I have made an exception for this one because so many variants of it have occurred in recent submissions.

For n>=2, a(n) = product of odd positive integers <=(n-1). - Jaroslav Krizek, Mar 21 2009

a(n) is, for n>=3, the number of way to choose floor((n-1)/2) disjoint pairs of items from n-1 items. It is then a fortiori the size of the conjugacy class of the reversal permutation [n-1,n-2,n-3,...,3,2,1]=(1 n-1)(2 n-2)(3 n-3)... in the symmetric group on n-1 elements. - Karl-Dieter Crisman, Nov 03 2009

Links

• G. C. Greubel, Table of n, a(n) for n = 0..800

Crossrefs

Cf. A006882, A055634.

Appears in A161736. - Johannes W. Meijer, Jun 18 2009

Programs

Mathematica

f[x_] := E^(x^2/2) + Sqrt[Pi/2]*Erfi[x/Sqrt[2]]; CoefficientList[ Series[f[x], {x, 0, 29}], x]*Range[0, 29]! (* Jean-François Alcover, Sep 25 2012, after Sergei N. Gladkovskii *)
Table[(n - 1 - Mod[n, 2])!!, {n, 0, 20}] (* Eric W. Weisstein, Dec 31 2017 *)
Table[((2 n + (-1)^n - 3)/2)!!, {n, 0, 20}] (* Eric W. Weisstein, Dec 31 2017 *)

PARI

a(n) = my(k = (2*n + (-1)^n - 3)/2); prod(i=0, (k-1)\2, k - 2*i) \\ Iain Fox, Dec 31 2017

Sage

def Gauss_factorial(N, n): return mul(j for j in (1..N) if gcd(j, n) == 1)
def A133221(n): return Gauss_factorial(n-1, 2)
[A133221(n) for n in (0..29)]  # Peter Luschny, Oct 01 2012

Formula

E.g.f.: x*U(0) where U(k)= 1 + (2*k+1)/(x - x^4/(x^3 + (2*k+2)*(2*k+3)/U(k+1))) ; (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Sep 25 2012

G.f.: 1+x*G(0), where G(k)= 1 + x*(2*k+1)/(1 - x/(x + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 07 2013

a(n) = (2*floor(n/2)-1)!! = (n-1-(n mod 2))!!. - Alois P. Heinz, Sep 24 2024