A174549 - OEIS

%I #40 Sep 08 2022 08:45:51

%S 3,30,840,45360,3991680,518918400,93405312000,22230464256000,

%T 6758061133824000,2554547108585472000,1175091669949317120000,

%U 646300418472124416000000,418802671169936621568000000,315777214062132212662272000000,274094621805930760590852096000000

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

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

%C Also the number of adjacency matrices for the n-helm graph. - _Eric W. Weisstein_, May 25 2017

%H Vincenzo Librandi, <a href="/A174549/b174549.txt">Table of n, a(n) for n = 1..100</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AdjacencyMatrixt.html">Adjacency Matrix</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HelmGraph.html">Helm Graph</a>.

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

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

%F 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)). Verified for n=1 to 4 on Wolfram Alpha. - _Jean-Claude Babois_, Oct 28 2014

%F Sum_{n>=1} (-1)^(n+1)/a(n) = sin(1)-cos(1) = (-1)*A143624. - _Amiram Eldar_, Apr 12 2021

%p A174549 := proc(n) (1+2*n)*(2*n-1)! ; end proc: # _R. J. Mathar_, Jan 13 2011

%t Table[(2*n - 1)! + (2*n)!, {n, 15}] (* _T. D. Noe_, Nov 21 2013 *)

%o (Magma) [Factorial(2*n-1) + Factorial(2*n): n in [1..15]]; // _Vincenzo Librandi_, Aug 04 2011

%o (PARI) a(n)=(2*n-1)!*(2*n+1) \\ _Charles R Greathouse IV_, Nov 21 2013

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

%K nonn,easy

%O 1,1

%A _Paul Curtz_, Mar 22 2010