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

2, 48, 3840, 645120, 185794560, 81749606400, 51011754393600, 42849873690624000, 46620662575398912000, 63777066403145711616000, 107145471557284795514880000, 216862434431944426122117120000

Offset: 0

Author

Benoit Cloitre, Feb 02 2002

Keywords

• nonn

Comments

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

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100
• Eric Weisstein's World of Mathematics, Riemann-Siegel Functions

Crossrefs

Cf. A000165.

Programs

Magma

[2^(n+1)*Factorial(n+1): n in [0..30 by 2]]; // Vincenzo Librandi, Feb 07 2014

Maple

for n from 0 to 30 by 2 do printf(`%d,`,2^(n+1)*(n+1)!) od: # James Sellers, Feb 11 2002

Mathematica

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 *)
Table[2^(n+1) (n+1)!,{n,0,30,2}] (* Harvey P. Dale, Feb 06 2014 *)

Formula

a(n) = A000165(2n+2) where A000165(k) are the double factorial numbers 2^k*k!=(2k)!!.

sqrt((1-cos(x))/2) = sum(n>=0, (-1)^n * x^(2*n+1) / a(n) ).

Sum_{n>=0} 1/a(n) = sinh(1/2) = (exp(1) - 1)/(2*exp(1/2)). - Ilya Gutkovskiy, Jun 08 2016

Extensions

More terms from James Sellers, Feb 11 2002