cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A099282 Decimal expansion of the cosine integral at 1.

Original entry on oeis.org

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

Views

Author

Robert G. Wilson v, Oct 08 2004

Keywords

Examples

			I=0.337403922900968134662646203889150769997578032585731894801318542...

Links

Crossrefs

Cf. A001620, A099281, A257535.

Programs

Formula

Equals -real(E_1(I)) = -(E_1(I)+E_1(-I))/2, where E_1(z) = Integral_{t>=1} exp(-t*z)/t dt is the exponential integral. - Stanislav Sykora, May 08 2015
From Amiram Eldar, Jun 25 2021: (Start)
Equals -Integral_{x>=1} cos(x)/x dx.
Equals gamma - Integral_{x=0..1} (1-cos(x))/x dx, where gamma = A001620.
Equals gamma + Sum_{k>=1} (-1)^k/((2*k)*(2*k)!). (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

Views

Author

Stanislav Sykora, Apr 28 2015

Keywords

Comments

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

Examples

			0.6247132564277136042899683778165717842862467449494411200160175...

Links

Crossrefs

Cf. A099281, A099282, A099285.

Programs

  • Mathematica
    RealDigits[Pi/2 - SinIntegral[1], 10, 105][[1]] (* Amiram Eldar, May 24 2023 *)
  • PARI
    a = imag(-eint1(I))

Formula

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

Views

Author

Wolfdieter Lang, Jan 23 2004

Keywords

Links

Crossrefs

Cf. A002674 (first column of A090438), A091033 (third column), A090438.
Cf. A001620, A049470, A073742, A073743, A099281, A099282, A099284.

Programs

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

Formula

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

Views

Author

Wolfdieter Lang, Jan 23 2004

Keywords

Crossrefs

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.

Programs

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

Formula

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

Views

Author

Wolfdieter Lang, Jan 23 2004

Keywords

Crossrefs

Cf. A091033 (third column of A090438), A091035 (fifth column), A090438.
Cf. A001620, A049469, A049470, A073742, A073743, A099281, A099282, A099283, A099284.

Programs

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

Formula

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

Views

Author

Wolfdieter Lang, Jan 23 2004

Keywords

Crossrefs

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.

Programs

  • Mathematica
    Table[Binomial[2n-2,4] (2n)!/6!,{n,3,20}] (* Harvey P. Dale, Jun 07 2021 *)
  • PARI
    a(n) = binomial(2*n-2, 4)*(2*n)!/6!; \\ Amiram Eldar, Nov 03 2022

Formula

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)

A101512 Decimal expansion of the sine integral at 2*Pi.

Original entry on oeis.org

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

Views

Author

Robert G. Wilson v, Jan 25 2005

Keywords

Comments

Si(Infinity)=Pi/2=~1.57079... (A019669).

Examples

			1.41815157613262845024578016229974942914245334929501627184272022969...

Crossrefs

Cf. A019669, A036792, A099281.

Programs

  • Mathematica
    RealDigits[ SinIntegral[2Pi], 10, 111] [[1]]

Formula

Equals Integral_{x=0..2*Pi} sin(x)/x dx.

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

Views

Author

Jean-François Alcover, Sep 04 2014

Keywords

Examples

			0.3433779615564270328325330038583124340012440194999075192...

Links

Crossrefs

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

Programs

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

Formula

(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).
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.