A010050 a(n) = (2n)!.

1, 2, 24, 720, 40320, 3628800, 479001600, 87178291200, 20922789888000, 6402373705728000, 2432902008176640000, 1124000727777607680000, 620448401733239439360000, 403291461126605635584000000, 304888344611713860501504000000, 265252859812191058636308480000000

Offset: 0

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

• nonn
• easy

Comments

Denominators in the expansion of cos(x): cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + x^8/8! - ...

Contribution from Peter Bala, Feb 21 2011: (Start)

We may compare the representation a(n) = Product_{k = 0..n-1} (n*(n+1)-k*(k+1)) with n! = Product_{k = 0..n-1} (n-k). Thus we may view a(n) as a generalized factorial function associated with the oblong numbers A002378. Cf. A000680.

The associated generalized binomial coefficients a(n)/(a(k)*a(n-k)) are triangle A086645, cf. A186432. (End)

Also, this sequence is the denominator of cosh(x) = (e^x + e^(-x))/2 = 1 + x^2/2! + x^4/4! + x^6/6! + ... - Mohammad K. Azarian, Jan 19 2012

Also (2n+1)-th derivative of arccoth(x) at x = 0. - Michel Lagneau, Aug 18 2012

Product of the partition parts of 2n+1 into exactly two positive integer parts, n > 0. Example: a(3) = 720, since 2(3)+1 = 7 has 3 partitions with exactly two positive integer parts: (6,1), (5,2), (4,3). Multiplying the parts in these partitions gives: 6! = 720. - Wesley Ivan Hurt, Jun 03 2013

Examples

			G.f. = 1 + 2*x + 24*x^2 + 720*x^3 + 40320*x^4 + 3628800*x^5 + ...
		

References

• John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 110.
• 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, chapters 32 and 33, equations 32:6:1 and 33:6:1 at pages 300 and 314.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• W. Dunham, Touring the calculus gallery, Amer. Math. Monthly, 112 (2005), 1-19.
• Alois Panholzer, Parking function varieties for combinatorial tree models, arXiv:2007.14676 [math.CO], 2020.
• Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
• Eric Weisstein's World of Mathematics, Hyperbolic Cosine.
• Index entries for related partition-counting sequences.

Crossrefs

Cf. A000142, A000165, A009445, A073743.

Bisection of A005359, |A012251|, A012254, A070734.

Programs

Magma

[Factorial(2*n): n in [0..15]]; // Vincenzo Librandi, Aug 21 2011

Maple

A010050 := proc(n) (2*n)! ;end proc: # R. J. Mathar, Feb 28 2011

Mathematica

a[n_]:=(2n)!; Array[a,16,0] (* Stefano Spezia, Jan 02 2025 *)

PARI

a(n)=(n*2)! \\ M. F. Hasler, Apr 22 2015

Sage

[stirling_number1(2*n+1,1) for n in range(0, 22)] # Zerinvary Lajos, Nov 26 2009

Formula

a(n) = 2^n*A000680(n).

E.g.f.: arctanh(x) = Sum_{k>=0} a(k) * x^(2*k+1)/ (2*k+1)!.

E.g.f.: 1/(1-x^2) = Sum_{k>=0} a(k) * x^(2*k) / (2*k)!. - Paul Barry, Sep 14 2004

D-finite with recurrence: a(n+1) = a(n)*(2*n+1)*(2*n+2) = a(n)*A002939(n-1). - Lekraj Beedassy, Apr 29 2005

a(n) = Product_{k = 1..n} (2*k*n-k*(k-1)). - Peter Bala, Feb 21 2011

G.f.: G(0) where G(k) = 1 + 2*x*(2*k+1)*(4*k+1)/(1 - 4*x*(k+1)*(4*k+3)/(4*x*(k+1)*(4*k+3) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 18 2012

a(n) = 2*A002674(n), n > 0. - Wesley Ivan Hurt, Jun 05 2013

From Ilya Gutkovskiy, Jan 20 2017: (Start)

a(n) ~ 2*sqrt(Pi)*4^n*n^(2*n+1/2)/exp(2*n).

Sum_{n>=0} 1/a(n) = cosh(1) = A073743. (End)

Extensions

Third line of data from M. F. Hasler, Apr 22 2015