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A010050 (2n)!. 57
1, 2, 24, 720, 40320, 3628800, 479001600, 87178291200, 20922789888000, 6402373705728000, 2432902008176640000, 1124000727777607680000, 620448401733239439360000, 403291461126605635584000000, 304888344611713860501504000000, 265252859812191058636308480000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

REFERENCES

W. Dunham, Touring the calculus gallery, Amer. Math. Monthly, 112 (2005), 1-19.

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

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.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math.CO/0606404, Jan 05, 2007

Eric Weisstein's World of Mathematics, Hyperbolic Cosine

Index entries for related partition-counting sequences

FORMULA

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

E.g.f. for sequence with alt. signs: arctan(x).

E.g.f. : 1/(1-x^2) (with interpolated zeros). - Paul Barry, Sep 14 2004

a(n+1) = a(n)*(2n+1)*(2n+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, 3-step). - Sergei N. Gladkovskii, Nov 18 2012

G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - (2*k+1)*(2*k+2)/(1-x/(x - 1/G(k+1) )); ( continued fraction, 3-step ). - Sergei N. Gladkovskii, Jan 15 2013

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

MAPLE

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

MATHEMATICA

s=1; lst={s}; Do[s+=(s*=n)*n; AppendTo[lst, s], {n, 1, 5!, 2}]; lst (*Vladimir Joseph Stephan Orlovsky, Nov 15 2008 *)

PROG

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

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

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

CROSSREFS

Cf. A000142, A000165, A009445.

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

Sequence in context: A188959 A093459 * A186246 A012161 A009724 A177771

Adjacent sequences:  A010047 A010048 A010049 * A010051 A010052 A010053

KEYWORD

nonn,easy

AUTHOR

Joe Keane (jgk(AT)jgk.org)

EXTENSIONS

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

STATUS

approved

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Last modified December 7 22:57 EST 2016. Contains 278899 sequences.