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A067626 a(n) = 2^(2n+1)*(2n+1)!.

%I A067626 #44 Jul 02 2025 16:02:01
%S A067626 2,48,3840,645120,185794560,81749606400,51011754393600,
%T A067626 42849873690624000,46620662575398912000,63777066403145711616000,
%U A067626 107145471557284795514880000,216862434431944426122117120000
%N A067626 a(n) = 2^(2n+1)*(2n+1)!.
%C A067626 a(n) equals the absolute value of the imaginary part of the determinant of the (4n+2) X (4n+2) matrix with i's along the superdiagonal (where i is the imaginary unit) and 2, 3, 4, ..., 4n+2 along the subdiagonal, and 0's everywhere else (see Mathematica code below). - _John M. Campbell_, Jun 04 2011
%C A067626 These are the denominators in the Taylor (Maclaurin) series expansion of sin(x) about the point x_0 = 0 and then evaluated at x = Pi/2.  Hence, Sum_{n>=0}(-1)^n*Pi^(2n+1)/a(n) = 1. - _Geoffrey Critzer_, Jun 08 2016
%H A067626 Vincenzo Librandi, <a href="/A067626/b067626.txt">Table of n, a(n) for n = 0..100</a>
%H A067626 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Riemann-SiegelFunctions.html">Riemann-Siegel Functions</a>
%F A067626 a(n) = A000165(2n+2) where A000165(k) are the double factorial numbers 2^k*k!=(2k)!!.
%F A067626 sqrt((1-cos(x))/2) = sum(n>=0, (-1)^n * x^(2*n+1) / a(n) ).
%F A067626 Sum_{n>=0} 1/a(n) = sinh(1/2) = (exp(1) - 1)/(2*exp(1/2)). - _Ilya Gutkovskiy_, Jun 08 2016
%p A067626 for n from 0 to 30 by 2 do printf(`%d,`,2^(n+1)*(n+1)!) od: # _James Sellers_, Feb 11 2002
%t A067626 Table[Abs[Im[Det[Array[KroneckerDelta[#1 + 1, #2]*I &, {4*n + 2, 4*n + 2}] + Array[KroneckerDelta[#1 - 1, #2]*#1 &, {4*n + 2, 4*n + 2}]]]], {n, 0, 20}] (* _John M. Campbell_, Jun 04 2011 *)
%t A067626 Table[2^(n+1) (n+1)!,{n,0,30,2}] (* _Harvey P. Dale_, Feb 06 2014 *)
%o A067626 (Magma) [2^(n+1)*Factorial(n+1): n in [0..30 by 2]]; // _Vincenzo Librandi_, Feb 07 2014
%Y A067626 Cf. A000165.
%K A067626 nonn
%O A067626 0,1
%A A067626 _Benoit Cloitre_, Feb 02 2002
%E A067626 More terms from _James Sellers_, Feb 11 2002