A174549 a(n) = (2*n-1)! + (2*n)!.

3, 30, 840, 45360, 3991680, 518918400, 93405312000, 22230464256000, 6758061133824000, 2554547108585472000, 1175091669949317120000, 646300418472124416000000, 418802671169936621568000000, 315777214062132212662272000000, 274094621805930760590852096000000

Offset: 1

Author

Paul Curtz, Mar 22 2010

Keywords

• nonn
• easy

Comments

x*cos(x) - sin(x) = Sum_{n>=1} (-1)^n/a(n) * x^(2*n+1). - James R. Buddenhagen, Nov 21 2013

Also the number of adjacency matrices for the n-helm graph. - Eric W. Weisstein, May 25 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..100
• Eric Weisstein's World of Mathematics, Adjacency Matrix.
• Eric Weisstein's World of Mathematics, Helm Graph.

Crossrefs

Cf. A001048, A000255, A068985, A143624, A165457.

Programs

Magma

[Factorial(2*n-1) + Factorial(2*n): n in [1..15]]; // Vincenzo Librandi, Aug 04 2011

Maple

A174549 := proc(n) (1+2*n)*(2*n-1)! ; end proc: # R. J. Mathar, Jan 13 2011

Mathematica

Table[(2*n - 1)! + (2*n)!, {n, 15}] (* T. D. Noe, Nov 21 2013 *)

PARI

a(n)=(2*n-1)!*(2*n+1) \\ Charles R Greathouse IV, Nov 21 2013

Formula

a(n) = A001048(2n) = (1+2n)*(2n-1)! = 3*A165457(n-1).

Sum_{n>=1} 1/a(n) = A068985 = 1/e = lim_{n->infinity} A000255(n-1)/A001048(n).

zeta(2*n+1) = Integral_{u=0..Pi/2} (sin(u)*log(sin(u))^(2*n+1)/(cos(u)^3))*(-2)^(2*n+1)/(n*a(n)) du. Verified for n=1 to 4 on Wolfram Alpha. - Jean-Claude Babois, Oct 28 2014

Sum_{n>=1} (-1)^(n+1)/a(n) = sin(1)-cos(1) = (-1)*A143624. - Amiram Eldar, Apr 12 2021