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%I #74 Jan 02 2025 12:45:13
%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 Isaac Newton, De analysi, 1669; reprinted in D. Whiteside, ed., The Mathematical Works of Isaac Newton, vol. 1, Johnson Reprint Co., 1964; see p. 20.
%D Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 32, equation 32:6:2 at page 301.
%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
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / (1 - 4*x)^(3/2). - _Ilya Gutkovskiy_, Jul 11 2021
%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 range(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 (Haskell)
%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.coefficient(x,4*n+2) for n in (0..15)] # _Peter Luschny_, Dec 14 2012
%Y Cf. A000142, A001147, A010050, A014481.
%K nonn,easy,changed
%O 0,2
%A _R. H. Hardin_, Joe Keane (jgk(AT)jgk.org)