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

1, 6, 120, 5040, 362880, 39916800, 6227020800, 1307674368000, 355687428096000, 121645100408832000, 51090942171709440000, 25852016738884976640000, 15511210043330985984000000, 10888869450418352160768000000, 8841761993739701954543616000000, 8222838654177922817725562880000000

Offset: 0

Author

R. H. Hardin, Joe Keane (jgk(AT)jgk.org)

Keywords

• nonn
• easy

Comments

Denominators in the expansion of sin(x):

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

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

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

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

Examples

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

References

• John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 111.
• H. B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan, NY, 1968, p. 88.
• 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.
• Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 32, equation 32:6:2 at page 301.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, et al., Enumeration of idempotents in diagram semigroups and algebras, arXiv preprint arXiv:1408.2021 [math.GR], 2014.
• W. Dunham, Touring the calculus gallery, Amer. Math. Monthly, 112 (2005), 1-19.
• Eric Weisstein's World of Mathematics, Hyperbolic Sine.

Crossrefs

Cf. A000142, A001147, A010050, A014481.

Programs

Haskell

a009445 n = product [1..2*n+1]  -- Reinhard Zumkeller, Dec 03 2011

Magma

[Factorial(2*n+1): n in [0..20]]; // Vincenzo Librandi, Oct 21 2011

Mathematica

Array[(2 # + 1)! &, 15] (* Robert G. Wilson v, Aug 08 2018 *)

PARI

a(n)=(n+n+1)! \\ Charles R Greathouse IV, Oct 20 2011

Sage

[stirling_number1(2*i,1) for i in range(1,22)] # Zerinvary Lajos, Jun 27 2008

Sage

T = taylor(sin(x^2), x, 0, 70)
[(-1)^n/T.coefficient(x,4*n+2) for n in (0..15)] # Peter Luschny, Dec 14 2012

Formula

a(n) = A014481(n) * A001147(n). - Reinhard Zumkeller, Dec 03 2011

Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / (1 - 4*x)^(3/2). - Ilya Gutkovskiy, Jul 11 2021