A073743 -id:A073743 - cp's OEIS frontend

A091032 Second column (k=3) of array A090438 ((4,2)-Stirling2) divided by 8.

1, 60, 5040, 604800, 99792000, 21794572800, 6102480384000, 2134124568576000, 912338253066240000, 468333636574003200000, 284372184127734743040000, 201645730563302817792000000, 165147853331345007771648000000

Offset: 2

Wolfdieter Lang, Jan 23 2004

John Campbell, Applications of Gosper’s nonlocal derangement identity, Maple Transactions, 5, 1, Article 16724, Feb. 2025.

Cf. A002674 (first column of A090438), A091033 (third column), A090438.

Cf. A001620, A049470, A073742, A073743, A099281, A099282, A099284.

a[n_] := (n - 1)*(2*n)!/4!; Array[a, 13, 2] (* Amiram Eldar, Nov 03 2022 *)

a(n) = (n-1)*(2*n)!/4!; \\ Amiram Eldar, Nov 03 2022

a(n) = A090438(n, 3)/8 = (n-1)*(2*n)!/4!
E.g.f.: (-3*hypergeom([1/2, 1], [], 4*x) + hypergeom([1, 3/2], [], 4*x) + 2)/(8*3!) (cf. A090438).
From Amiram Eldar, Nov 03 2022: (Start)
Sum_{n>=2} 1/a(n) = 60 - 24*Gamma - 24*cosh(1) + 24*CoshIntegral(1) - 24*sinh(1).
Sum_{n>=2} (-1)^n/a(n) = -12 + 24*gamma - 24*cos(1) - 24*CosIntegral(1) + 24*SinIntegral(1). (End)
a(n+1) = Sum_{j = 1..n} (-1)^(n+j) * j^(2*n+2) * binomial(2*n, n-j) (Campbell, Eq. 17). - Peter Bala, Mar 30 2025

A091034 Fourth column (k=5) of array A090438 ((4,2)-Stirling2) divided by 24.

Original entry on oeis.org

1, 280, 70560, 19958400, 6659452800, 2644408166400, 1244905998336000, 689322235650048000, 444916954745303040000, 331767548149023866880000, 283424276847308960563200000, 275246422218908346286080000000

Offset: 3

Wolfdieter Lang, Jan 23 2004

Cf. A091033 (third column of A090438), A091035 (fifth column), A090438.

Cf. A001620, A049469, A049470, A073742, A073743, A099281, A099282, A099283, A099284.

a[n_] := (n - 1)*(n - 2)*(2*n - 3)*(2*n)!/(5!*(3!)^2); Array[a, 12, 3] (* Amiram Eldar, Nov 03 2022 *)

a(n) = (n-1)*(n-2)*(2*n-3)*(2*n)!/(5!*(3!)^2); \\ Amiram Eldar, Nov 03 2022

a(n) = A090438(n, 5)/24, n>=3.
a(n) = (n-1)*(n-2)*(2*n-3)*(2*n)!/(5!*(3!)^2), n>=3.
E.g.f.: (Sum_{p=2..5} (((-1)^(p+1))*binomial(5, p)*hypergeom([(p-1)/2, p/2], [], 4*x)) + 4)/(5!*4!) (cf. A090438).
From Amiram Eldar, Nov 03 2022: (Start)
Sum_{n>=3} 1/a(n) = 2010 - 4680*Gamma + 1800*cosh(1) + 4680*CoshIntegral(1) - 2520*sinh(1) - 2880*SinhIntegral(1).
Sum_{n>=3} (-1)^(n+1)/a(n) = -2010 - 3960*gamma + 3240*cos(1) + 3960*CosIntegral(1) - 1800*sin(1) + 2880*SinIntegral(1). (End)

A091035 Fifth column (k=6) of array A090438 ((4,2)-Stirling2).

Original entry on oeis.org

1, 840, 352800, 139708800, 59935075200, 29088489830400, 16183777978368000, 10339833534750720000, 7563588230670151680000, 6303583414831453470720000, 5951909813793488171827200000, 6330667711034891964579840000000

Offset: 3

Wolfdieter Lang, Jan 23 2004

Cf. A091034 (fourth column of A090438 divided by 24), A091036 (sixth column divided by 48), A053134, A090438.

Cf. A001620, A049469, A049470, A073742, A073743, A099281, A099282, A099283, A099284.

Table[Binomial[2n-2,4] (2n)!/6!,{n,3,20}] (* Harvey P. Dale, Jun 07 2021 *)

a(n) = binomial(2*n-2, 4)*(2*n)!/6!; \\ Amiram Eldar, Nov 03 2022

a(n) = A090438(n, 6), n>=3.
a(n) = binomial(2*n-2, 4)*(2*n)!/6! = A053134(n-3)*(2*n)!/6!, n>=3.
E.g.f.: (Sum_{p=2..6} (((-1)^p)*binomial(6, p)*hypergeom([(p-1)/2, p/2], [], 4*x)) - 5)/6! (cf. A090438).
From Amiram Eldar, Nov 03 2022: (Start)
Sum_{n>=3} 1/a(n) = -594 + 1800*Gamma - 1008*cosh(1) - 1800*CoshIntegral(1) + 912*sinh(1) + 1464*SinhIntegral(1).
Sum_{n>=3} (-1)^(n+1)/a(n) = 1554 + 1080*gamma - 1248*cos(1) - 1080*CosIntegral(1) + 240*sin(1) - 1416*SinIntegral(1). (End)

A334400 Decimal expansion of cosh(e).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 26 2020

			(e^e + e^(-e))/2 = 7.6101251386622883634186102301133791...

Cf. A001113, A073226, A073230, A073743, A085660, A334399.

Mathematica
```
RealDigits[Cosh[E], 10, 110] [[1]]
```

Equals Sum_{k>=0} e^(2*k)/(2*k)!.

A365927 Decimal expansion of arccosh(e).

Original entry on oeis.org

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

Offset: 1

Kritsada Moomuang, Oct 15 2023

			1.65745445415307727259382874228053...

Cf. A001113, A073743, A085660, A188739, A334400, A366599.

Mathematica
```
RealDigits[ArcCosh[E], 10, 100] [[1]]
```

acosh(exp(1)) \\ Amiram Eldar, Oct 18 2023

Equals log(A188739). - Amiram Eldar, Oct 18 2023

A196932 Decimal expansion of sinh(1)*cosh(1).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Oct 23 2011

Also decimal expansion of sinh(2)/2.

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 14 2019

			sinh(1)*cosh(1) = sinh(2)/2 = 1.8134302039235093838341...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for transcendental numbers

Cf. A001113, A073742, A073743.

Sinh(2)/2; // G. C. Greubel, Jun 10 2018

RealDigits[Sinh[2]/2, 10, 100][[1]] (* G. C. Greubel, Jun 10 2018 *)

sinh(2)/2 \\ Charles R Greathouse IV, Apr 21 2016

Equals Product_{k>=1} 1 + 4/(k * Pi)^2. - Amiram Eldar, Jul 16 2020

a(25)-a(72) from John W. Layman, Oct 24 2011

A322544 a(n) is the reciprocal of the coefficient of x^n in the power series defined by ((1+2x)exp(x) + 3exp(-x) - 4)/ (4x^2).

Original entry on oeis.org

1, 6, 8, 60, 180, 1680, 8064, 90720, 604800, 7983360, 68428800, 1037836800, 10897286400, 186810624000, 2324754432000, 44460928512000, 640237370572800, 13516122267648000, 221172909834240000, 5109094217170944000, 93666727314800640000, 2350183339898634240000, 47726800133326110720000

Offset: 0

Pierre-Alain Sallard, Dec 14 2018

nonn

Muniru A Asiru, Table of n, a(n) for n = 0..445

Cf. A060593 (even bisection, shifted), A028242 (denominator minus 1), A030451 (denominator, shifted), A107991 (Expansion of a similar function), A073743.

List([0..25],n->(4*Factorial(n+2))/(2*n+5+3*(-1)^n)); # Muniru A Asiru, Dec 20 2018

a:=n->factorial(n+2)/(3*floor(n/2)-n+2): seq(a(n),n=0..25); # Muniru A Asiru, Dec 20 2018

Table[4*Factorial[n + 2]/(2*n + 5 + 3*(-1)^n), {n, 0, 25}]
(* or *)
Function[x, 1/x] /@
CoefficientList[Series[(Exp[x]/4 + 3/4*Exp[-x] + x/2*Exp[x] - 1)/x^2, {x, 0, 20}], x]

a(n)={(4*(n+2)!)/(5 + 3*(-1)^n + 2*n)} \\ Andrew Howroyd, Dec 14 2018

my(x='x + O('x^30)); Vec(apply(x->1/x, ((1+2*x)*exp(x) + 3*exp(-x) - 4)/ (4*x^2))) \\ Michel Marcus, Dec 19 2018

a(n) = (n+2)!/(3*floor(n/2)-n+2).
a(n) = (4*(n+2)!)/(2n+5+3*(-1)^n).
a(n) = 4/([x^n]((exp(x)*(1+2x)+3*exp(-x)-4)/x^2)).
a(n) = (n+2)!/(A028242(n)+1).
a(n) = (n+2)!/A030451(n+1).
a(n) ~ sqrt(Pi/2)/72*exp(-n)*n^(n-1/2)*(1705 - 264*n + 288*n^2). - Stefano Spezia, Aug 11 2025
Sum_{n>=0} 1/a(n) = 3*cosh(1)/2 - 1. - Amiram Eldar, Aug 15 2025

A334366 Decimal expansion of Sum_{k>=0} 1/(4*k)!!.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 24 2020

This constant is transcendental.

			1/(2^0*0!) + 1/(2^2*2!) + 1/(2^4*4!) + ... = 1.1276259652063807852...

Index entries for transcendental numbers

Cf. A019774, A067624, A073743, A143280, A160327, A201505, A334367.

Mathematica
```
RealDigits[Cosh[1/2], 10, 110] [[1]]
```
PARI
```
cosh(1/2) \\ Michel Marcus, Apr 25 2020
```

Equals cosh(1/2).

Equals Product_{k>=0} 1 + 1/((2*k+1)*Pi)^2. - Amiram Eldar, Jul 16 2020

A349804 Decimal expansion of cosh(1) - cos(1).

Original entry on oeis.org

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

Offset: 1

Christoph B. Kassir, Dec 11 2021

			1.00277832894710406107696901331408507...

Michael Ian Shamos, Shamos's catalog of the real numbers, (2011).

Cf. A073743, A049470, A334364, A334365.

RealDigits[Cosh[1] - Cos[1], 10, 100][[1]] (* Amiram Eldar, Dec 11 2021 *)

PARI
```
cosh(1) - cos(1)
```

Equals 2 * Sum_{k>=0} 1/(4*k+2)! = 2 * A334364. - Amiram Eldar, Dec 11 2021

A372338 Decimal expansion of Sum_{k>=0} (k+2) / ((k+1)*(2k)!).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 28 2024

			2.8073217524723591352868580804341...

Cf. A073742, A073743, A372339.

s = Sum[(k + 2)/((k + 1) (2 k)!), {k, 0, Infinity}]
d = N[s, 100]
First[RealDigits[d]]

Equals -4/e + 2 + 3*cosh(1) - 2*sinh(1).

Equals 2*2F3(1,3; 1/2,2,2; 1/4). - R. J. Mathar, Aug 02 2024

cp's OEIS Frontend

A091032 Second column (k=3) of array A090438 ((4,2)-Stirling2) divided by 8.

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A091034 Fourth column (k=5) of array A090438 ((4,2)-Stirling2) divided by 24.

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A091035 Fifth column (k=6) of array A090438 ((4,2)-Stirling2).

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A334400 Decimal expansion of cosh(e).

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A365927 Decimal expansion of arccosh(e).

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A196932 Decimal expansion of sinh(1)*cosh(1).

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A322544 a(n) is the reciprocal of the coefficient of x^n in the power series defined by ((1+2x)*exp(x) + 3*exp(-x) - 4)/ (4x^2).

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A334366 Decimal expansion of Sum_{k>=0} 1/(4*k)!!.

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A322544 a(n) is the reciprocal of the coefficient of x^n in the power series defined by ((1+2x)exp(x) + 3exp(-x) - 4)/ (4x^2).