This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A133221 #36 Sep 24 2024 09:55:11
%S A133221 1,1,1,1,3,3,15,15,105,105,945,945,10395,10395,135135,135135,2027025,
%T A133221 2027025,34459425,34459425,654729075,654729075,13749310575,
%U A133221 13749310575,316234143225,316234143225,7905853580625,7905853580625,213458046676875,213458046676875
%N A133221 A001147 with each term repeated.
%C A133221 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.
%C A133221 For n>=2, a(n) = product of odd positive integers <=(n-1). - _Jaroslav Krizek_, Mar 21 2009
%C A133221 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
%H A133221 G. C. Greubel, <a href="/A133221/b133221.txt">Table of n, a(n) for n = 0..800</a>
%F A133221 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
%F A133221 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
%F A133221 a(n) = (2*floor(n/2)-1)!! = (n-1-(n mod 2))!!. - _Alois P. Heinz_, Sep 24 2024
%t A133221 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_ *)
%t A133221 Table[(n - 1 - Mod[n, 2])!!, {n, 0, 20}] (* _Eric W. Weisstein_, Dec 31 2017 *)
%t A133221 Table[((2 n + (-1)^n - 3)/2)!!, {n, 0, 20}] (* _Eric W. Weisstein_, Dec 31 2017 *)
%o A133221 (Sage)
%o A133221 def Gauss_factorial(N, n): return mul(j for j in (1..N) if gcd(j, n) == 1)
%o A133221 def A133221(n): return Gauss_factorial(n-1, 2)
%o A133221 [A133221(n) for n in (0..29)] # _Peter Luschny_, Oct 01 2012
%o A133221 (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
%Y A133221 Cf. A006882, A055634.
%Y A133221 Appears in A161736. - _Johannes W. Meijer_, Jun 18 2009
%K A133221 nonn
%O A133221 0,5
%A A133221 _N. J. A. Sloane_, Oct 13 2007