A051396 -id:A051396 - cp's OEIS frontend

A073743 Decimal expansion of cosh(1).

1, 5, 4, 3, 0, 8, 0, 6, 3, 4, 8, 1, 5, 2, 4, 3, 7, 7, 8, 4, 7, 7, 9, 0, 5, 6, 2, 0, 7, 5, 7, 0, 6, 1, 6, 8, 2, 6, 0, 1, 5, 2, 9, 1, 1, 2, 3, 6, 5, 8, 6, 3, 7, 0, 4, 7, 3, 7, 4, 0, 2, 2, 1, 4, 7, 1, 0, 7, 6, 9, 0, 6, 3, 0, 4, 9, 2, 2, 3, 6, 9, 8, 9, 6, 4, 2, 6, 4, 7, 2, 6, 4, 3, 5, 5, 4, 3, 0, 3, 5, 5, 8, 7, 0, 4

Offset: 1

Rick L. Shepherd, Aug 07 2002

Also decimal expansion of cos(i). - N. J. A. Sloane, Feb 12 2010
cosh(x) = (e^x + e^(-x))/2.
Equals Sum_{n>=0} 1/A010050(n). See Gradsteyn-Ryzhik (0.245.5). - R. J. Mathar, Oct 27 2012
By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 14 2019

			1.54308063481524377847790562075...

S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 218.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, equation 2:5:6 at page 20.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Hyperbolic Cosine.
Eric Weisstein's World of Mathematics, Hyperbolic Functions.
Eric Weisstein's World of Mathematics, Factorial Sums.
Index entries for transcendental numbers.

Cf. A068118 (continued fraction), A073742, A073744, A073745, A073746, A073747, A049470, A137204.

Digits:=100: evalf(cosh(1)); # Wesley Ivan Hurt, Nov 18 2014

RealDigits[Cosh[1],10,120][[1]] (* Harvey P. Dale, Aug 03 2014 *)

PARI
```
cosh(1)
```

Continued fraction representation: cosh(1) = 1 + 1/(2 - 2/(13 - 12/(31 - ... - (2*n - 4)*(2*n - 5)/((4*n^2 - 10*n + 7) - ... )))). See A051396 for proof. Cf. A049470 (cos(1)) and A073742 (sinh(1)). - Peter Bala, Sep 05 2016
Equals Product_{k>=0} 1 + 4/((2*k+1)*Pi)^2. - Amiram Eldar, Jul 16 2020
Equals 1/A073746 = A137204/2. - Hugo Pfoertner, Dec 27 2024

A087350 a(n) = Sum_{k=0..n} (3n)!/(3k)!.

Original entry on oeis.org

1, 7, 841, 423865, 559501801, 1527439916731, 7478345832314977, 59677199741873516461, 724719913665311983902385, 12718834484826225317486856751, 309830808050366848733979830454361, 10142621332336809160155563729753961697, 434509897877308904421064350182659719099481

Offset: 0

Vladeta Jovovic, Oct 20 2003

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..100

Cf. A000522, A051396.

a(n)={sum(k=0, n, (3*n)!/(3*k)!)} \\ Andrew Howroyd, Jan 27 2020

a(n) = floor((3*n)!*C) where C = 1/3*exp(1)+2/3*exp(-1/2)*cos(1/2*3^(1/2)) = 1.16805831337591852551625692...

Terms a(11) and beyond from Andrew Howroyd, Jan 27 2020

A337725 a(n) = (3n+1)! Sum_{k=0..n} 1 / (3*k+1)!.

Original entry on oeis.org

1, 25, 5251, 3780721, 6487717237, 21798729916321, 126737815733490295, 1171057417377450325801, 16160592359808814496053801, 317652603424402057734433512457, 8567090714356123497097671830965291, 307592825008242258039794809418977808065

Offset: 0

Ilya Gutkovskiy, Sep 17 2020

nonn

Cf. A000522, A051396, A051397, A087350, A100089, A143820, A330044, A337726, A337727, A337728, A337729, A337730.

Table[(3 n + 1)! Sum[1/(3 k + 1)!, {k, 0, n}], {n, 0, 11}]
Table[(3 n + 1)! SeriesCoefficient[(Exp[3 x/2] - 2 Sin[Pi/6 - Sqrt[3] x/2])/(3 Exp[x/2] (1 - x^3)), {x, 0, 3 n + 1}], {n, 0, 11}]
Table[Floor[(Exp[3/2] + 2 Sin[(3 Sqrt[3] - Pi)/6])/(3 Sqrt[Exp[1]]) (3 n + 1)!], {n, 0, 11}]

a(n) = (3*n+1)!*sum(k=0, n, 1/(3*k+1)!); \\ Michel Marcus, Sep 17 2020

E.g.f.: (exp(3*x/2) - 2 * sin(Pi/6 - sqrt(3)*x/2)) / (3*exp(x/2) * (1 - x^3)) = x + 25*x^4/4! + 5251*x^7/7! + 3780721*x^10/10! + ...

a(n) = floor(c * (3*n+1)!), where c = (exp(3/2) + 2 * sin((3 * sqrt(3) - Pi) / 6))/(3 * sqrt(exp(1))) = A143820.

A337727 a(n) = (4n)! Sum_{k=0..n} 1 / (4*k)!.

Original entry on oeis.org

1, 25, 42001, 498971881, 21795091762081, 2534333270094778681, 646315807872650838343345, 317599587988620621961919733001, 274101148417699141578015206369183041, 387502275541069630431671657548241448722521, 849931991080760484603611346800010863970028660561

Offset: 0

Ilya Gutkovskiy, Sep 17 2020

nonn

Cf. A000522, A051396, A051397, A087350, A100733, A330045, A332890, A337725, A337726, A337728, A337729, A337730.

Table[(4 n)! Sum[1/(4 k)!, {k, 0, n}], {n, 0, 10}]
Table[(4 n)! SeriesCoefficient[(1/2) (Cos[x] + Cosh[x])/(1 - x^4), {x, 0, 4 n}], {n, 0, 10}]
Table[Floor[(1/2) (Cos[1] + Cosh[1]) (4 n)!], {n, 0, 10}]

a(n) = (4*n)!*sum(k=0, n, 1/(4*k)!); \\ Michel Marcus, Sep 17 2020

E.g.f.: (1/2) * (cos(x) + cosh(x)) / (1 - x^4) = 1 + 25*x^4/4! + 42001*x^8/8! + 498971881*x^12/12! + ...

a(n) = floor(c * (4*n)!), where c = (cos(1) + cosh(1)) / 2 = A332890.

A337728 a(n) = (4n+1)! Sum_{k=0..n} 1 / (4*k+1)!.

Original entry on oeis.org

1, 121, 365905, 6278929801, 358652470233121, 51516840824285500441, 15640512874253077933887601, 8915467710633236496186345872425, 8755702529258688898174686554391144001, 13878488965077362598718732163634314533105081, 33731389859841228248933904149069928786421237268881

Offset: 0

Ilya Gutkovskiy, Sep 17 2020

nonn

Cf. A000522, A051396, A051397, A087350, A330045, A334363, A337725, A337726, A337727, A337729, A337730.

Table[(4 n + 1)! Sum[1/(4 k + 1)!, {k, 0, n}], {n, 0, 10}]
Table[(4 n + 1)! SeriesCoefficient[(1/2) (Sin[x] + Sinh[x])/(1 - x^4), {x, 0, 4 n + 1}], {n, 0, 10}]
Table[Floor[(1/2) (Sin[1] + Sinh[1]) (4 n + 1)!], {n, 0, 10}]

a(n) = (4*n+1)!*sum(k=0, n, 1/(4*k+1)!); \\ Michel Marcus, Sep 17 2020

E.g.f.: (1/2) * (sin(x) + sinh(x)) / (1 - x^4) = x + 121*x^5/5! + 365905*x^9/9! + 6278929801*x^13/13! + ...

a(n) = floor(c * (4*n+1)!), where c = (sin(1) + sinh(1)) / 2 = A334363.

A337726 a(n) = (3n+2)! Sum_{k=0..n} 1 / (3*k+2)!.

Original entry on oeis.org

1, 61, 20497, 20292031, 44317795705, 180816606476401, 1236785588298582841, 13142083661260741268467, 205016505115667563788085201, 4494781858155895668489979946725, 133764708098719455094261803214536001, 5252940087036713001551661012234828759271

Offset: 0

Ilya Gutkovskiy, Sep 17 2020

nonn

Cf. A000522, A051396, A051397, A087350, A100043, A143821, A330044, A337725, A337727, A337728, A337729, A337730.

Table[(3 n + 2)! Sum[1/(3 k + 2)!, {k, 0, n}], {n, 0, 11}]
Table[(3 n + 2)! SeriesCoefficient[(Exp[3 x/2] - 2 Sin[Sqrt[3] x/2 + Pi/6])/(3 Exp[x/2] (1 - x^3)), {x, 0, 3 n + 2}], {n, 0, 11}]
Table[Floor[(Exp[3/2] - 2 Sin[(3 Sqrt[3] + Pi)/6])/(3 Sqrt[Exp[1]]) (3 n + 2)!], {n, 0, 11}]

a(n) = (3*n+2)!*sum(k=0, n, 1/(3*k+2)!); \\ Michel Marcus, Sep 17 2020

E.g.f.: (exp(3*x/2) - 2 * sin(sqrt(3)*x/2 + Pi/6)) / (3*exp(x/2) * (1 - x^3)) = x^2/2! + 61*x^5/5! + 20497*x^8/8! + 20292031*x^11/11! + ...

a(n) = floor(c * (3*n+2)!), where c = (exp(3/2) - 2 * sin((3 * sqrt(3) + Pi) / 6))/(3 * sqrt(exp(1))) = A143821.

A337729 a(n) = (4n+2)! Sum_{k=0..n} 1 / (4*k+2)!.

Original entry on oeis.org

1, 361, 1819441, 43710250585, 3210080802962401, 563561785768079119561, 202205968733586788098486801, 132994909755454702268136738753721, 148026526435655214537290625514621562305, 262237873172349351865682580536682974917045801, 704454843460345510903820429747302209179158476142321

Offset: 0

Ilya Gutkovskiy, Sep 17 2020

nonn

Cf. A000522, A051396, A051397, A087350, A330045, A334364, A337725, A337726, A337727, A337728, A337730.

Table[(4 n + 2)! Sum[1/(4 k + 2)!, {k, 0, n}], {n, 0, 10}]
Table[(4 n + 2)! SeriesCoefficient[(1/2) (Cosh[x] - Cos[x])/(1 - x^4), {x, 0, 4 n + 2}], {n, 0, 10}]
Table[Floor[(1/2) (Cosh[1] - Cos[1]) (4 n + 2)!], {n, 0, 10}]

a(n) = (4*n+2)!*sum(k=0, n, 1/(4*k+2)!); \\ Michel Marcus, Sep 17 2020

E.g.f.: (1/2) * (cosh(x) - cos(x)) / (1 - x^4) = x^2/2! + 361*x^6/6! + 1819441*x^10/10! + 43710250585*x^14/14! + ...

a(n) = floor(c * (4*n+2)!), where c = (cosh(1) - cos(1)) / 2 = A334364.

A337730 a(n) = (4n+3)! Sum_{k=0..n} 1 / (4*k+3)!.

Original entry on oeis.org

1, 841, 6660721, 218205219961, 20298322381652065, 4313799472548696853801, 1816972337837511114820981201, 1372104830641374893468212163747161, 1724241814377177346127894133451232399041, 3403694723384093133512770088891935585284510985

Offset: 0

Ilya Gutkovskiy, Sep 17 2020

nonn

Cf. A000522, A051396, A051397, A087350, A330045, A334365, A337725, A337726, A337727, A337728, A337729.

Table[(4 n + 3)! Sum[1/(4 k + 3)!, {k, 0, n}], {n, 0, 9}]
Table[(4 n + 3)! SeriesCoefficient[(1/2) (Sinh[x] - Sin[x])/(1 - x^4), {x, 0, 4 n + 3}], {n, 0, 9}]
Table[Floor[(1/2) (Sinh[1] - Sin[1]) (4 n + 3)!], {n, 0, 9}]

a(n) = (4*n+3)!*sum(k=0, n, 1/(4*k+3)!); \\ Michel Marcus, Sep 17 2020

E.g.f.: (1/2) * (sinh(x) - sin(x)) / (1 - x^4) = x^3/3! + 841*x^7/7! + 6660721*x^11/11! + 218205219961*x^15/15! + ...

a(n) = floor(c * (4*n+3)!), where c = (sinh(1) - sin(1)) / 2 = A334365.

A275651 a(n) = (2n)!Sum_{k = 0..n} (-1)^k/(2*k)!.

Original entry on oeis.org

1, 1, 13, 389, 21785, 1960649, 258805669, 47102631757, 11304631621681, 3459217276234385, 1314502564969066301, 607300185015708631061, 335229702128671164345673

Offset: 0

Peter Bala, Sep 02 2016

Compare with the derangement numbers A000166(n) := n!*sum_{k = 0..n} (-1)^k/k! and also A074790.

Seiichi Manyama, Table of n, a(n) for n = 0..225

Cf. A000166, A049470 (cos(1)), A051396, A073743, A074790.

A275651 := proc(n) option remember; if (n = 0) then 1 else 2*n*(2*n - 1)*A275651(n-1)+(-1)^n end if; end proc:
seq(A275651(n), n = 0..20);

Table[(2 n)!*Sum[(-1)^k/(2 k)!, {k, 0, n}], {n, 12}] (* Michael De Vlieger, Sep 04 2016 *)

a(n) ~ (2*n)!*cos(1).
E.g.f. for the aerated sequence: cos(x)/(1 - x^2) = 1 + x^2/2! + 13*x^4/4! + 389*x^6/6! + ....
Recurrence equations:
a(n) = 2*n*(2*n - 1)*a(n-1) + (-1)^n with a(0) = 1.
a(n) = (4*n^2 - 2*n - 1)*a(n - 1) + (2*n - 2)*(2*n - 3)*a(n - 2) with a(0) = 1, a(1) = 1.
The latter recurrence is also satisfied by the sequence b(n) := (2*n)! with b(0) = 1, b(1) = 2. This leads to the continued fraction representation a(n) = (2*n )!*( 1/(1 + 1/(1 + 2/(11 + 12/(29 + ... + (2*n - 2)*(2*n - 3)/(4*n^2 - 2*n - 1) )))) ) for n >= 3. Taking the limit gives the continued fraction representation cos(1) = A049470 = 1/(1 + 1/(1 + 2/(11 + 12/(29 + ... + (2*n - 2)*(2*n - 3)/((4*n^2 - 2*n - 1) + ... ))))). Cf. A073743.

A119828 Number triangle T(n,k)=(2n)!/(2k)!.

Original entry on oeis.org

1, 2, 1, 24, 12, 1, 720, 360, 30, 1, 40320, 20160, 1680, 56, 1, 3628800, 1814400, 151200, 5040, 90, 1, 479001600, 239500800, 19958400, 665280, 11880, 132, 1, 87178291200, 43589145600, 3632428800, 121080960, 2162160, 24024, 182, 1

Offset: 0

Paul Barry, May 25 2006

Row sums are A051396(n+1). Diagonal sums are A119829. Inverse is the bi-diagonal array A119830. E.g.f. cosh(x*y)/(1-y*x^2) produces an aerated version.

			Triangle begins as:
        1,
        2,       1,
       24,      12,      1,
      720,     360,     30,    1,
    40320,   20160,   1680,   56,  1,
  3628800, 1814400, 151200, 5040, 90, 1

Flatten[Table[(2n)!/(2k)!, {n, 0, 7}, {k, 0, n}]] (* James C. McMahon, Sep 23 2024 *)

Number triangle T(n,k)=[k<=n](2n)!/(2k)!

cp's OEIS Frontend

A073743 Decimal expansion of cosh(1).

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A087350 a(n) = Sum_{k=0..n} (3*n)!/(3*k)!.

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A337725 a(n) = (3*n+1)! * Sum_{k=0..n} 1 / (3*k+1)!.

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A337727 a(n) = (4*n)! * Sum_{k=0..n} 1 / (4*k)!.

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A337728 a(n) = (4*n+1)! * Sum_{k=0..n} 1 / (4*k+1)!.

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A337726 a(n) = (3*n+2)! * Sum_{k=0..n} 1 / (3*k+2)!.

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A337729 a(n) = (4*n+2)! * Sum_{k=0..n} 1 / (4*k+2)!.

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A337730 a(n) = (4*n+3)! * Sum_{k=0..n} 1 / (4*k+3)!.

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A275651 a(n) = (2*n)!*Sum_{k = 0..n} (-1)^k/(2*k)!.

A087350 a(n) = Sum_{k=0..n} (3n)!/(3k)!.

A337725 a(n) = (3n+1)! Sum_{k=0..n} 1 / (3*k+1)!.

A337727 a(n) = (4n)! Sum_{k=0..n} 1 / (4*k)!.

A337728 a(n) = (4n+1)! Sum_{k=0..n} 1 / (4*k+1)!.

A337726 a(n) = (3n+2)! Sum_{k=0..n} 1 / (3*k+2)!.

A337729 a(n) = (4n+2)! Sum_{k=0..n} 1 / (4*k+2)!.

A337730 a(n) = (4n+3)! Sum_{k=0..n} 1 / (4*k+3)!.

A275651 a(n) = (2n)!Sum_{k = 0..n} (-1)^k/(2*k)!.