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 1-4 of 4 results.

A258232 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k) dx.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, May 24 2015

Keywords

Examples

			0.3684125359314336523213165973278510150142413039288199683036158...

References

  • József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 424.

Links

Crossrefs

Cf. A000120, A010815, A029931, A242168, A258229, A258230, A258408.
Cf. A258406 (m=2), A258407 (m=3), A258404 (m=4), A258405 (m=5).

Programs

  • Maple
    evalf(8*sqrt(3/23)*Pi*sinh(sqrt(23)*Pi/6)/(2*cosh(sqrt(23)*Pi/3)-1), 123);
evalf(Sum((-1)^n/((3*n-1)*n/2 + 1), n=-infinity..infinity), 123);
  • Mathematica
    RealDigits[N[8*Sqrt[3/23]*Pi*Sinh[Sqrt[23]*Pi/6] / (2*Cosh[Sqrt[23]*Pi/3]-1),120]][[1]]
  • PARI
    8*Pi*sqrt(3/23) * sinh(sqrt(23)*Pi/6) / (2*cosh(sqrt(23)*Pi/3) - 1) \\ Michel Marcus, Nov 28 2018

Formula

Equals 8*Pi*sqrt(3/23) * sinh(sqrt(23)*Pi/6) / (2*cosh(sqrt(23)*Pi/3) - 1).
From Amiram Eldar, Feb 04 2024: (Start)
Equals 2 * Sum_{k=-oo..oo} (-1)^k/(3*k^2 + k + 2).
Equals Sum_{k>=0} (-1)^A000120(k)/(A029931(k)+1) (Borwein and Borwein, 1992). (End)

A258191 Numerator of Integral_{x=0..1} Product_{k=1..n} x^k*(1-x^k) dx.

Original entry on oeis.org

1, 11, 293, 487, 129952159, 13084761625, 8277192566411, 576489266167410341, 2154341459717480222819111, 1562388737113054944319018297, 4507453407946726622146977923716952747, 46170199302621715634236277404186409941, 20107953791404084220109855379873778475523352268948164789
Offset: 1

Views

Author

Vaclav Kotesovec, May 23 2015

Keywords

Comments

Limit n->infinity (a(n)/A258192(n))^(1/n) = 0.185155...
The limit is equal to 0.1851552893223595946473132111979542852738... = 1/5.400871904118154152466091119104270052029... (see A258234). - Vaclav Kotesovec, May 24 2015

Examples

			Product_{k=1..n} x^k*(1-x^k)
n=1 x - x^2
n=2 x^3 - x^4 - x^5 + x^6
n=3 x^6 - x^7 - x^8 + x^10 + x^11 - x^12
Integral Product_{k=1..n} x^k*(1-x^k) dx
n=1 x^2/2 - x^3/3
n=2 x^4/4 - x^5/5 - x^6/6 + x^7/7
n=3 x^7/7 - x^8/8 - x^9/9 + x^11/11 + x^12/12 - x^13/13
For Integral_{x=0..1} set x=1
n=1 1/2 - 1/3 = 1/6, a(1)=1
n=2 1/4 - 1/5 - 1/6 + 1/7 = 11/420, a(2)=11
n=3 1/7 - 1/8 - 1/9 + 1/11 + 1/12 - 1/13 = 293/72072, a(3)=293

Links

Crossrefs

Cf. A258192, A258229, A258230, A258234.

Programs

  • Mathematica
    nmax=15; p=1; Table[p=Expand[p*x^n*(1-x^n)]; Total[CoefficientList[p,x]/Range[1,Exponent[p,x]+1]], {n,1,nmax}] // Numerator

A258192 Denominator of Integral_{x=0..1} Product_{k=1..n} x^k*(1-x^k) dx.

Original entry on oeis.org

6, 420, 72072, 760760, 1266697832400, 783333734619744, 3002950101013562700, 1253414030788528596187200, 27809824888100301666382826331840, 118802724769051077369996224554510800, 2005396188718644499811084404372455793370133120
Offset: 1

Views

Author

Vaclav Kotesovec, May 23 2015

Keywords

Comments

Limit n->infinity (A258191(n)/a(n))^(1/n) = 0.185155...
The limit is equal to 0.1851552893223595946473132111979542852738... = 1/5.400871904118154152466091119104270052029... (see A258234). - Vaclav Kotesovec, May 24 2015

Examples

			Product_{k=1..n} x^k*(1-x^k)
n=1 x - x^2
n=2 x^3 - x^4 - x^5 + x^6
n=3 x^6 - x^7 - x^8 + x^10 + x^11 - x^12
Integral Product_{k=1..n} x^k*(1-x^k) dx
n=1 x^2/2 - x^3/3
n=2 x^4/4 - x^5/5 - x^6/6 + x^7/7
n=3 x^7/7 - x^8/8 - x^9/9 + x^11/11 + x^12/12 - x^13/13
For Integral_{x=0..1} set x=1
n=1 1/2 - 1/3 = 1/6, a(1)=6
n=2 1/4 - 1/5 - 1/6 + 1/7 = 11/420, a(2)=420
n=3 1/7 - 1/8 - 1/9 + 1/11 + 1/12 - 1/13 = 293/72072, a(3)=72072

Links

Crossrefs

Cf. A258191, A258229, A258230, A258234.

Programs

  • Mathematica
    nmax=15; p=1; Table[p=Expand[p*x^n*(1-x^n)]; Total[CoefficientList[p,x]/Range[1,Exponent[p,x]+1]], {n,1,nmax}] // Denominator

A258230 Denominator of Integral_{x=0..1} Product_{k=1..n} (1-x^k) dx.

Original entry on oeis.org

2, 12, 105, 495, 55440, 340340, 1012647636, 12304749600, 5920545668637600, 1098951951860282520, 1572101004939647757775200, 2051717579526635495717258016, 244523633377266327241371614400, 32818916025992059215981780272862841200
Offset: 1

Views

Author

Vaclav Kotesovec, May 24 2015

Keywords

Comments

Limit n->infinity A258229(n) / a(n) = limit n->infinity Integral_{x=0..1} Product_{k=1..n} (1-x^k) dx = 8*sqrt(3/23)*Pi*sinh(sqrt(23)*Pi/6) / (2*cosh(sqrt(23)*Pi/3)-1) = A258232 = 0.368412535931433652321316597327851...

Examples

			Product_{k=1..n} (1-x^k)
n=1 1 - x
n=2 1 - x - x^2 + x^3
n=3 1 - x - x^2 + x^4 + x^5 - x^6
Integral Product_{k=1..n} (1-x^k) dx
n=1 x - x^2/2
n=2 x - x^2/2 - x^3/3 + x^4/4
n=3 x - x^2/2 - x^3/3 + x^5/5 + x^6/6 - x^7/7
For Integral_{x=0..1} set x=1
n=1 1 - 1/2 = 1/2, a(1) = 2
n=2 1 - 1/2 - 1/3 + 1/4 = 5/12, a(2) = 12
n=3 1 - 1/2 - 1/3 + 1/5 + 1/6 - 1/7 = 41/105, a(3) = 105

Links

Crossrefs

Cf. A258229, A258191, A258192, A258232.

Programs

  • Mathematica
    nmax=15; p=1; Table[p=Expand[p*(1-x^n)]; Total[CoefficientList[p,x]/Range[1,Exponent[p,x]+1]], {n,1,nmax}] // Denominator
Showing 1-4 of 4 results.

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