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1, 2, 24, 720, 40320, 3628800, 479001600, 87178291200, 20922789888000, 6402373705728000, 2432902008176640000, 1124000727777607680000, 620448401733239439360000, 403291461126605635584000000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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!+ . . . .---[From Mohammad K. Azarian, January 19, 2012].
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REFERENCES
| W. Dunham, Touring the calculus gallery, Amer. Math. Monthly, 112 (2005), 1-19.
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. B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan, NY, 1968, p. 88.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Hyperbolic Cosine
Index entries for related partition-counting sequences
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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 (pbarry(AT)wit.ie), Sep 14 2004
a(n+1) = a(n)*(2n+1)*(2n+2) = a(n)*A002939(n-1). - Lekraj Beedassy (blekraj(AT)yahoo.com), Apr 29 2005
a(n) = product {k = 1..n} (2*k*n-k*(k-1)) - Peter Bala, Feb 21 2011
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MAPLE
| A010050 := proc(n) (2*n)! ; end proc: # R. J. Mathar, Feb 28 2011
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MATHEMATICA
| s=1; lst={s}; Do[s+=(s*=n)*n; AppendTo[lst, s], {n, 1, 5!, 2}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 15 2008]
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PROG
| (Other) sage: [stirling_number1(2*n+1, 1) for n in xrange(0, 22)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 26 2009]
(MAGMA)[Factorial(2*n): n in [0..15]]; // Vincenzo Librandi, Aug 21 2011
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CROSSREFS
| Cf. A000142, A000165, A009445.
Bisection of A005359, |A012251|, A012254, A070734.
Sequence in context: A119699 A188959 A093459 * A012161 A009724 A177771
Adjacent sequences: A010047 A010048 A010049 * A010051 A010052 A010053
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KEYWORD
| nonn,easy
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AUTHOR
| Joe Keane (jgk(AT)jgk.org)
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