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A163085 Product of first n swinging factorials (A056040).

%I A163085 #17 Jun 26 2013 10:55:11
%S A163085 1,1,2,12,72,2160,43200,6048000,423360000,266716800000,67212633600000,
%T A163085 186313420339200000,172153600393420800000,2067909047925770649600000,
%U A163085 7097063852481244869427200000
%N A163085 Product of first n swinging factorials (A056040).
%C A163085 With the definition of the Hankel transform as given by Luschny (see link) which uniquely determines the original sequence (provided that all determinants are not zero) this is also 1/ the Hankel determinant of 1/(n+1) (assuming (0,0)-based matrices).
%C A163085 a(2*n-1) is 1/determinant of the Hilbert matrix H(n) (A005249).
%C A163085 a(2*n) = A067689(n). - _Peter Luschny_, Sep 18 2012
%H A163085 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SequenceTransformations#Hankel">SequenceTransformations</a>
%p A163085 a := proc(n) local i; mul(A056040(i),i=0..n) end;
%t A163085 a[0] = 1; a[n_] := a[n] = a[n-1]*n!/Floor[n/2]!^2; Table[a[n], {n, 0, 14}] (* _Jean-François Alcover_, Jun 26 2013 *)
%o A163085 (Sage)
%o A163085 def A056040(n):
%o A163085     swing = lambda n: factorial(n)/factorial(n//2)^2
%o A163085     return mul(swing(i) for i in (0..n))
%o A163085 [A056040(i) for i in (0..14)] # _Peter Luschny_, Sep 18 2012
%Y A163085 Cf. A056040, A163086, A055462, A000178.
%K A163085 nonn
%O A163085 0,3
%A A163085 _Peter Luschny_, Jul 21 2009