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 A009445 a(n) = (2*n+1)!. 30

%I

%S 1,6,120,5040,362880,39916800,6227020800,1307674368000,

%T 355687428096000,121645100408832000,51090942171709440000,

%U 25852016738884976640000,15511210043330985984000000,10888869450418352160768000000,8841761993739701954543616000000,8222838654177922817725562880000000

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

%C Denominators in the expansion of sin(x):

%C sin(x) = x - x^3/3! + x^5/5! - x^7/7! + x^9/9! - ...

%C Denominators in the expansion of sinc(x) = sin(x)/x:

%C sinc x = sin(x)/x = 1 - x^2/3! + x^4/5! - x^6/7! + x^8/9! - ... - _Daniel Forgues_, Oct 20 2011

%C The terms of this sequence are the denominators of sinh(x) = (e^x-e^(-x))/2 = x + x^3/3! + x^5/5! + x^7/7! + .... - _Mohammad K. Azarian_, Jan 19 2012

%D H. B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan, NY, 1968, p. 88.

%D I. Newton, De analysi, 1669; reprinted in D. Whiteside, ed., The Mathematical Works of Isaac Newton, vol. 1, Johnson Reprint Co., 1964; see p. 20.

%H Vincenzo Librandi, <a href="/A009445/b009445.txt">Table of n, a(n) for n = 0..200</a>

%H I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, et al., <a href="http://arxiv.org/abs/1408.2021">Enumeration of idempotents in diagram semigroups and algebras</a>, arXiv preprint arXiv:1408.2021 [math.GR], 2014.

%H W. Dunham, <a href="http://www.jstor.org/stable/30037380">Touring the calculus gallery</a>, Amer. Math. Monthly, 112 (2005), 1-19.

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

%F a(n) = A014481(n) * A001147(n). - _Reinhard Zumkeller_, Dec 03 2011

%e G.f. = 1 + 6*x + 120*x^2 + 5040*x^3 + 362880*x^4 + 39916800*x^5 + ...

%t Array[(2 # + 1)! &, 15] (* _Robert G. Wilson v_, Aug 08 2018 *)

%o (Sage) [stirling_number1(2*i,1) for i in xrange(1,22)] # _Zerinvary Lajos_, Jun 27 2008

%o (PARI) a(n)=(n+n+1)! \\ _Charles R Greathouse IV_, Oct 20 2011

%o (MAGMA) [Factorial(2*n+1): n in [0..20]]; // _Vincenzo Librandi_, Oct 21 2011

%o a009445 n = product [1..2*n+1] -- _Reinhard Zumkeller_, Dec 03 2011

%o (Sage)

%o T = taylor(sin(x^2), x, 0, 70)

%o [(-1)^n/T.coeff(x,4*n+2) for n in (0..15)] # _Peter Luschny_, Dec 14 2012

%Y Cf. A010050, A000142.

%K nonn,easy

%O 0,2

%A _R. H. Hardin_, Joe Keane (jgk(AT)jgk.org)

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Last modified March 20 05:34 EDT 2019. Contains 321344 sequences. (Running on oeis4.)