A099282 -id:A099282 - cp's OEIS frontend

A099281 Decimal expansion of the sine integral at 1.

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

Offset: 0

Robert G. Wilson v, Oct 08 2004

			0.946083070367183014941353313823179657812337954738111790471454773...

HP 82480A Math Pac Owner's Manual For the HP-71, Section 11: Numerical Integration, Page 118.

Joseph K. Horn and O. Praem, Numerical Integration .

Cf. A099282, A257535.

RealDigits[ SinIntegral[1], 10, 105][[1]]

intnum(x=0,1,sinc(x)) \\ Charles R Greathouse IV, May 08 2015

Pi/2-imag(eint1(-I)) \\ Charles R Greathouse IV, May 08 2015

From Amiram Eldar, Aug 21 2020: (Start)
Equals Integral_{x=0..1} sin(x)/x dx.
Equals -Integral_{x=0..1} log(x)*cos(x) dx.
Equals Sum_{k>=0} (-1)^k/((2*k+1)*(2*k+1)!). (End)

A257535 Decimal expansion of the imaginary part of -E_1(i), i being the imaginary unit.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Apr 28 2015

E_1(z) = Integral_{t>=1}(exp(-t*z)/t) is the exponential integral.

			0.6247132564277136042899683778165717842862467449494411200160175...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Wikipedia, Exponential integral.

Cf. A099281, A099282, A099285.

RealDigits[Pi/2 - SinIntegral[1], 10, 105][[1]] (* Amiram Eldar, May 24 2023 *)

PARI
```
a = imag(-eint1(I))
```

Equals imag(E_1(-i)).

Equals (Pi/2) - A099281.

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

Original entry on oeis.org

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

A091033 Third column (k=4) of array A090438 ((4,2)-Stirling2).

Original entry on oeis.org

1, 180, 25200, 4233600, 898128000, 239740300800, 79332244992000, 32011868528640000, 15509750302126080000, 8898339094906060800000, 5971815866682429603840000, 4637851802955964809216000000

Offset: 2

Wolfdieter Lang, Jan 23 2004

Cf. A091032 (second column of A090438 divided by 8), A091034 (fourth column divided by 24), A000384, A090438.

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

a[n_] := (n-1)*(2*n-3)*(2*n)!/4!; Array[a, 12, 2] (* Amiram Eldar, Nov 03 2022 *)

a(n) = (n-1)*(2*n-3)*(2*n)!/4!; \\ Amiram Eldar, Nov 03 2022

a(n) = A090438(n, 4), n>=2.
a(n) = (n-1)*(2*n-3)*(2*n)!/4! = binomial(2*(n-1), 2)*(2*n)!/4! = A000384(n-1)*(2*n)!/4!, n>=2.
E.g.f.: (6*hypergeom([1/2, 1], [], 4*x) - 4*hypergeom([1, 3/2], [], 4*x) + hypergeom([3/2, 2], [], 4*x) -3)/4! (cf. A090438).
From Amiram Eldar, Nov 03 2022: (Start)
Sum_{n>=2} 1/a(n) = -20 + 24*Gamma - 16*CoshIntegral(1) + 16*sinh(1) + 8*SinhIntegral(1).
Sum_{n>=2} (-1)^n/a(n) = 4 - 24*gamma + 16*cos(1) + 24*CosIntegral(1) - 16*sin(1) + 8*SinIntegral(1). (End)

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)

A062779 a(n) = 2n(2*n)!.

Original entry on oeis.org

0, 4, 96, 4320, 322560, 36288000, 5748019200, 1220496076800, 334764638208000, 115242726703104000, 48658040163532800000, 24728016011107368960000, 14890761641597746544640000, 10485577989291746525184000000

Offset: 0

Jason Earls, Jul 18 2001

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 38, equation 38:6:2 at page 364.

Wikipedia, Cosine integral.
Wikipedia, Hyperbolic cosine integral.

Cf. A001563, A001620, A010050, A099282, A099284.

a[n_] := 2*n*(2*n)!; Array[a, 14, 0] (* Amiram Eldar, Feb 14 2021 *)

PARI
```
for(n=0,22,print((2*n)*(2*n)!))
```

From Amiram Eldar, Feb 14 2021: (Start)
a(n) = A001563(2*n) = 2*n*A010050(n).
Sum_{n>=1} 1/a(n) = Chi(1) - gamma = A099284 - A001620, where Chi(x) is the hyperbolic cosine integral
Sum_{n>=1} (-1)^(n+1)/a(n) = gamma - Ci(1) = A001620 - A099282, where Ci(x) is the cosine integral. (End)

A246820 Decimal expansion of integral_{0..infinity} x*exp(-x)/(1+x^2) dx.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 04 2014

			0.3433779615564270328325330038583124340012440194999075192...

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 52.

Cf. A073447, A073449, A099281, A099282, A224518.

RealDigits[(1/2)*Sin[1]*(Pi - 2*SinIntegral[1]) - Cos[1]*CosIntegral[1], 10, 104] // First

(1/2)*sin(1)*(Pi - 2*Si(1)) - cos(1)*Ci(1), where Si is the sine integral function and Ci the cosine integral function.
Also equals -integral_{0..1} log(x)/(1+log(x)^2) dx.
Also equals cot(1)*A224518 + Ci(1)*csc(1).

A257176 Decimal expansion of Integral_{x=0..1} Integral_{y=0..x} sin(x*y) dy dx.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Apr 17 2015

One of the examples in Mathematica's help file under "Integrate".

			0.1199058710002823629719329430966258305222906516770958520022243462243686983763...

Cf. A001620, A099282.

RealDigits[Integrate[Sin[x y], {x, 0, 1}, {y, 0, x}], 10, 111][[1]]

Equals (A001620 - A099282)/2.

cp's OEIS Frontend

A099281 Decimal expansion of the sine integral at 1.

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A257535 Decimal expansion of the imaginary part of -E_1(i), i being the imaginary unit.

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

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A091033 Third column (k=4) of array A090438 ((4,2)-Stirling2).

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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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A062779 a(n) = 2*n*(2*n)!.

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A246820 Decimal expansion of integral_{0..infinity} x*exp(-x)/(1+x^2) dx.

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A062779 a(n) = 2n(2*n)!.