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Product of first n swinging factorials (A056040).
9

%I #17 Jun 26 2013 10:55:11

%S 1,1,2,12,72,2160,43200,6048000,423360000,266716800000,67212633600000,

%T 186313420339200000,172153600393420800000,2067909047925770649600000,

%U 7097063852481244869427200000

%N Product of first n swinging factorials (A056040).

%C 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 a(2*n-1) is 1/determinant of the Hilbert matrix H(n) (A005249).

%C a(2*n) = A067689(n). - _Peter Luschny_, Sep 18 2012

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SequenceTransformations#Hankel">SequenceTransformations</a>

%p a := proc(n) local i; mul(A056040(i),i=0..n) end;

%t 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 (Sage)

%o def A056040(n):

%o swing = lambda n: factorial(n)/factorial(n//2)^2

%o return mul(swing(i) for i in (0..n))

%o [A056040(i) for i in (0..14)] # _Peter Luschny_, Sep 18 2012

%Y Cf. A056040, A163086, A055462, A000178.

%K nonn

%O 0,3

%A _Peter Luschny_, Jul 21 2009