A012254 -id:A012254 - cp's OEIS frontend

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

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

Offset: 0

Joe Keane (jgk(AT)jgk.org)

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

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

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.

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.

Cf. A000142, A000165, A009445, A073743.

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

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

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

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

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

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

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)

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

A296679 Expansion of e.g.f. arcsinh(arctanh(x)) (odd powers only).

Original entry on oeis.org

1, 1, 13, 341, 18649, 1599849, 205524837, 36391450941, 8546308276401, 2564025898856913, 957697868873929149, 435619128300038521893, 237104370189582892175241, 152148421079949399306125625, 113672892845152570858515803925, 97820056722556900357454981990925

Offset: 0

Ilya Gutkovskiy, Dec 18 2017

nonn

			arcsinh(arctanh(x)) = x/1! + x^3/3! + 13*x^5/5! + 341*x^7/7! + 18649*x^9/9! + 1599849*x^11/11! + ...

Cf. A001818, A003706, A010050, A012254, A096717, A096718, A296464, A296466, A296677.

nmax = 16; Table[(CoefficientList[Series[ArcSinh[ArcTanh[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
nmax = 16; Table[(CoefficientList[Series[Log[(Log[1 + x] - Log[1 - x])/2 + Sqrt[1 + (Log[1 + x] - Log[1 - x])^2/4]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

E.g.f.: arcsin(arctan(x)) (odd powers only, absolute values).

E.g.f.: log((log(1 + x) - log(1 - x))/2 + sqrt(1 + (log(1 + x) - log(1 - x))^2/4)) (odd powers only).

A012091 cos(arcsin(arctan(x)))=1-1/2!x^2+5/4!x^4-109/6!x^6+4521/8!x^8...

Original entry on oeis.org

1, -1, 5, -109, 4521, -330681, 36468717, -5721101541, 1205289090513, -328897528901361, 112769846051251797, -47472161654530776285, 24068680135210975714425, -14466929567049590650593705

Offset: 0

Patrick Demichel (patrick.demichel(AT)hp.com)

sign

Cf. A012254.

With[{nn=30},Take[CoefficientList[Series[Cos[ArcSin[ArcTan[x]]],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Feb 07 2015 *)

a[n]:=if n=0 then 1 else (2*n)!*sum(((2*k)!*binomial(2*k-2,k-1)*(-1)^(n+k+1)*sum((2^(i+1)*stirling1(i,2*k)*binomial(2*n-1,i-1))/i!, i, 2*k, 2*n))/(k*2^(4*k)), k, 1, 2*n); makelist(a[n], n, 0, 13); /* Vladimir Kruchinin, Oct 08 2012 */

a(n) = (2*n)!*sum(k=1..2*n, ((2*k)!*binomial(2*k-2,k-1)*(-1)^(n+k+1)*sum(i=2*k..2*n, (2^(i+1)*stirling1(i,2*k)*binomial(2*n-1,i-1))/i!))/(k*2^(4*k))) with n>0, a(0)=1. [Vladimir Kruchinin, Oct 08 2012]

cp's OEIS Frontend

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

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A296679 Expansion of e.g.f. arcsinh(arctanh(x)) (odd powers only).

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A012091 cos(arcsin(arctan(x)))=1-1/2!x^2+5/4!x^4-109/6!x^6+4521/8!x^8...

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

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Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A296679 Expansion of e.g.f. arcsinh(arctanh(x)) (odd powers only).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A012091 cos(arcsin(arctan(x)))=1-1/2!*x^2+5/4!*x^4-109/6!*x^6+4521/8!*x^8...

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Mathematica

Maxima

Formula

A012091 cos(arcsin(arctan(x)))=1-1/2!x^2+5/4!x^4-109/6!x^6+4521/8!x^8...