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%I #102 Sep 08 2022 08:44:37
%S 1,2,24,720,40320,3628800,479001600,87178291200,20922789888000,
%T 6402373705728000,2432902008176640000,1124000727777607680000,
%U 620448401733239439360000,403291461126605635584000000,304888344611713860501504000000,265252859812191058636308480000000
%N a(n) = (2n)!.
%C Denominators in the expansion of cos(x): cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + x^8/8! - ...
%C Contribution from _Peter Bala_, Feb 21 2011: (Start)
%C 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.
%C The associated generalized binomial coefficients a(n)/(a(k)*a(n-k)) are triangle A086645, cf. A186432. (End)
%C 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
%C Also (2n+1)-th derivative of arccoth(x) at x = 0. - _Michel Lagneau_, Aug 18 2012
%C 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
%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.
%H Vincenzo Librandi, <a href="/A010050/b010050.txt">Table of n, a(n) for n = 0..200</a>
%H W. Dunham, <a href="http://www.jstor.org/stable/30037380">Touring the calculus gallery</a>, Amer. Math. Monthly, 112 (2005), 1-19.
%H Alois Panholzer, <a href="https://arxiv.org/abs/2007.14676">Parking function varieties for combinatorial tree models</a>, arXiv:2007.14676 [math.CO], 2020.
%H Robert A. Proctor, <a href="http://arxiv.org/abs/math/0606404">Let's Expand Rota's Twelvefold Way For Counting Partitions!</a>, arXiv:math/0606404 [math.CO], 2006-2007.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HyperbolicCosine.html">Hyperbolic Cosine</a>
%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F a(n) = 2^n*A000680(n).
%F E.g.f.: arctanh(x) = Sum_{k>=0} a(k) * x^(2*k+1)/ (2*k+1)!.
%F E.g.f.: 1/(1-x^2) = Sum_{k>=0} a(k) * x^(2*k) / (2*k)!. - _Paul Barry_, Sep 14 2004
%F 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
%F a(n) = Product_{k = 1..n} (2*k*n-k*(k-1)). - _Peter Bala_, Feb 21 2011
%F 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
%F a(n) = 2*A002674(n), n > 0. - _Wesley Ivan Hurt_, Jun 05 2013
%F From _Ilya Gutkovskiy_, Jan 20 2017: (Start)
%F a(n) ~ 2*sqrt(Pi)*4^n*n^(2*n+1/2)/exp(2*n).
%F Sum_{n>=0} 1/a(n) = cosh(1) = A073743. (End)
%e G.f. = 1 + 2*x + 24*x^2 + 720*x^3 + 40320*x^4 + 3628800*x^5 + ...
%p A010050 := proc(n) (2*n)! ;end proc: # _R. J. Mathar_, Feb 28 2011
%o (Sage) [stirling_number1(2*n+1,1) for n in range(0, 22)] # _Zerinvary Lajos_, Nov 26 2009
%o (Magma)[Factorial(2*n): n in [0..15]]; // _Vincenzo Librandi_, Aug 21 2011
%o (PARI) a(n)=(n*2)! \\ _M. F. Hasler_, Apr 22 2015
%Y Cf. A000142, A000165, A009445, A073743.
%Y Bisection of A005359, |A012251|, A012254, A070734.
%K nonn,easy
%O 0,2
%A Joe Keane (jgk(AT)jgk.org)
%E Third line of data from _M. F. Hasler_, Apr 22 2015
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