A049469 -id:A049469 - cp's OEIS frontend

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

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)

A114940 Decimal expansion of the infinite sum Sum_{k>=1} sin(k)/k!.

Original entry on oeis.org

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

Offset: 1

Stefan Steinerberger, Feb 21 2006

This is the imaginary part of exp(exp(i)), i the imaginary unit, where the real part is 1+A114941. - R. J. Mathar, Apr 11 2024

			1.279883001373022493908...

Cf. A001113, A049469, A049470, A114941.

Sum[N[Sin[i], 400]/i!, {i, 1, 300}] (* This is accurate to 300 digits. *)

suminf(k=1, sin(k)/k!) \\ Michel Marcus, Jul 19 2020

From Amiram Eldar, Jul 19 2020: (Start)
Equals e^cos(1) * sin(sin(1)).
Equals sin(sin(1)) * (cosh(cos(1)) + sinh(cos(1))).
Equals (-i)*(e^(e^i) - e^(e^(-i)))/2. (End)

A117017 Decimal expansion of (sine of 1 radian)^(1/2).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 15 2006

			0.917317275978108081904271835360260308316831382531234230407306783565563168854...

Cf. A049469.

RealDigits[Sqrt[Sin[1]],10,120][[1]] (* Harvey P. Dale, Sep 29 2023 *)

A117017 = Sum_{n>=0} A008991(n)/A008992(n). - Johannes W. Meijer, Feb 10 2013

Offset corrected by Mohammad K. Azarian, Dec 11 2008

A222131 Decimal expansion of the imaginary part of Pi^i, where i=sqrt(-1).

Original entry on oeis.org

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

Offset: 0

Bruno Berselli, Feb 08 2013

			0.910598499212614707060044514236877474514929053377520207196164279559...

Vincenzo Librandi, Table of n, a(n) for n = 0..5000

Cf. A053510, A049469 (imaginary part of e^i), A222130 (real part of Pi^i).

 RealDigits[Im[Pi^I], 10, 90][[1]] (* or *) RealDigits[Sin[Log[Pi]], 10, 90][[1]]

fpprec:90; ev(bfloat(imagpart(%pi^%i)));

Equals sin(log(Pi)) = (Pi^i-1/Pi^i)/(2*i).

A280189 Version of sexagesimal expansion of sine of one degree given by the Persian mathematician Al-Kashi in the 15th Century.

Original entry on oeis.org

1, 2, 49, 43, 11, 14, 44, 16, 26, 17

Offset: 1

Mohammad K. Azarian, Dec 28 2016

The fifteenth century Persian mathematician Jamshid Al-Kashi was the first to calculate the value of sine of one degree correct to ten sexagesimal places (17 decimal digits) in his Risala al-Watar wa'l Jaib.

Mohammad K. Azarian, A Study of Risala al-Watar wa'l Jaib ("The Treatise on the Chord and Sine"), Forum Geometricorum, Volume 15 (2015) 229-242. Mathematical Reviews, MR 3418854 (Reviewed), Zentralblatt MATH, Zbl 1328.01015.

Cf. A019810, A019812, A049469, A110937, A280188.

A334378 Decimal expansion of Sum_{k>=0} 1/((2*k+1)!)^2.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 25 2020

			1/1!^2 + 1/3!^2 + 1/5!^2 + 1/7!^2 + ... = 1.027847261597415799692...
Continued fraction: 1 + 1/(36 - 36/(401 - 400/(1765 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (2*n*(2*n + 1))^2 for n >= 1. - _Peter Bala_, Feb 22 2024

Index entries for sequences related to Bessel functions or polynomials

Cf. A009445, A049469, A070910, A073742, A091681, A197036, A334379.

RealDigits[(BesselI[0, 2] - BesselJ[0, 2])/2, 10, 110] [[1]]

suminf(k=0, 1/((2*k+1)!)^2) \\ Michel Marcus, Apr 26 2020

(besseli(0,2) - besselj(0,2))/2 \\ Michel Marcus, Apr 26 2020

Equals (BesselI(0,2) - BesselJ(0,2))/2.

A068116 Continued fraction expansion for sin(1).

Original entry on oeis.org

0, 1, 5, 3, 4, 19, 2, 2, 2, 2, 7, 2, 2, 1, 136, 3, 20, 3, 1, 3, 2, 1, 1, 1, 1, 10, 11, 2, 9, 54, 3, 1, 16, 1, 1, 1, 1, 1, 4, 4, 1, 2, 1, 1, 3, 11, 1, 1, 25, 1, 1, 14, 3, 1, 2, 1, 2, 2, 2, 2, 1, 2, 9, 6, 1, 4, 4, 1, 5, 1, 14, 6, 2, 12, 1, 1, 1, 1, 2, 11, 15, 152

Offset: 0

Benoit Cloitre, Mar 13 2002

G. C. Greubel, Table of n, a(n) for n = 0..9999

Cf. A049469 (decimal expansion), A068115.

SetDefaultRealField(RealField(100)); ContinuedFraction(Sin(1)); // G. C. Greubel, Nov 30 2018

ContinuedFraction[Sin[1],90] (* Harvey P. Dale, Sep 16 2011 *)

default(realprecision, 100); contfrac(sin(1)) \\ G. C. Greubel, Nov 30 2018

continued_fraction_list(sin(1), nterms=100) # G. C. Greubel, Nov 30 2018

Offset changed by Andrew Howroyd, Aug 05 2024

A068878 Denominators of nonzero terms in the expansion of sin(x)+exp(x)-1-2*x.

Original entry on oeis.org

2, 24, 60, 720, 40320, 181440, 3628800, 479001600, 3113510400, 87178291200, 20922789888000, 177843714048000, 6402373705728000, 2432902008176640000, 25545471085854720000, 1124000727777607680000, 620448401733239439360000, 7755605021665492992000000, 403291461126605635584000000

Offset: 1

Benoit Cloitre, Mar 29 2002

Cf. A001113, A049469, A348374.

Select[CoefficientList[Series[Sin[x] + Exp[x] - 1 - 2 x, {x, 0, 25}], x], # != 0 &] // Denominator
(* second program: *)
a[n_] := Switch[Mod[n, 3], 1, ((4*n+2)/3)!, 2, ((4*n+4)/3)!, 0, ((4*n+3)/3)!/2]; Array[a, 17] (* Amiram Eldar, May 05 2025 *)

a(3k+1)=(4k+2)!, a(3k+2)=(4k+4)!, a(3k+3)=(4k+5)!/2.

Sum_{n>=1} 1/a(n) = e + sin(1) - 3 = A001113 + A049469 - 3 = A348374 - 3. - Amiram Eldar, May 05 2025

Name clarified by Sean A. Irvine, Mar 19 2024

More terms from Amiram Eldar, May 05 2025

A085662 Decimal expansion of sin(sin(1)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jul 15 2003

			0.74562414166555788889315107043038379205029164661536...

Jens Blanck, Exact real arithmetic systems: results of competition, pp. 389-393 of J. Blanck et al., eds., Computability and Complexity in Analysis (CCA 2000), Lect. Notes Computer Science, Springer-Verlag, 2001; alternative link.

Cf. A049469.

RealDigits[Sin[Sin[1]], 10, 120][[1]] (* Amiram Eldar, Jun 08 2023 *)

sin(sin(1)) \\ Charles R Greathouse IV, Mar 25 2014

A114941 Decimal expansion of the infinite sum Sum_{k>=1} cos(k)/k!.

Original entry on oeis.org

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

Offset: 0

Stefan Steinerberger, Feb 21 2006

			0.143835643791640325906...

Cf. A001113, A049469, A049470, A114940.

RealDigits[Sum[N[Cos[i],400]/i!, {i,1,300}]][[1]] (*which is accurate to 300 digits*) (* corrected by Harvey P. Dale, Nov 29 2011 *)

suminf(k=1, cos(k)/k!) \\ Michel Marcus, Jul 19 2020

From Amiram Eldar, Jul 19 2020: (Start)
Equals e^cos(1) * cos(sin(1)) - 1.
Equals cos(sin(1)) * (cosh(cos(1)) + sinh(cos(1))) - 1.
Equals (e^(e^i) + e^(e^(-i)))/2 - 1. (End)

cp's OEIS Frontend

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

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A114940 Decimal expansion of the infinite sum Sum_{k>=1} sin(k)/k!.

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Mathematica

PARI

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A117017 Decimal expansion of (sine of 1 radian)^(1/2).

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Mathematica

Formula

Extensions

A222131 Decimal expansion of the imaginary part of Pi^i, where i=sqrt(-1).

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Mathematica

Maxima

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A280189 Version of sexagesimal expansion of sine of one degree given by the Persian mathematician Al-Kashi in the 15th Century.

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A334378 Decimal expansion of Sum_{k>=0} 1/((2*k+1)!)^2.

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PARI

PARI

Formula

A068116 Continued fraction expansion for sin(1).

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Magma

Mathematica

PARI

Sage

Extensions

A068878 Denominators of nonzero terms in the expansion of sin(x)+exp(x)-1-2*x.

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