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 1-5 of 5 results.

A263030 Decimal expansion of a constant related to A262876 and A262946 (negated).

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Oct 08 2015

Keywords

Examples

			-0.18870819197952853237641009864920797359211446726842922150941743378232...

Crossrefs

Cf. A262876, A262877, A262946, A262947, A263031, A075700, A084448, A263406, A263415.

Programs

  • Mathematica
    NIntegrate[1/x*(Exp[-2*x]/(1 - Exp[-3*x])^2 - 1/(9*x^2) - 1/(9*x) + Exp[-x]/36), {x, 0, Infinity}, WorkingPrecision -> 120, MaxRecursion -> 100, PrecisionGoal -> 110]

Formula

Integral_{x=0..infinity} 1/x*(exp(-2*x)/(1 - exp(-3*x))^2 - 1/(9*x^2) - 1/(9*x) + exp(-x)/36) dx.
exp(3*(A263030+A263031)) = A^2 * Gamma(1/3) / (3^(11/12) * exp(1/6) * sqrt(2*Pi)), where A = A074962 is the Glaisher-Kinkelin constant.

A263405 Expansion of Product_{k>=1} 1/(1-x^(3*k+1))^k.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 2, 1, 0, 3, 2, 1, 4, 6, 2, 6, 10, 6, 8, 20, 14, 13, 30, 29, 22, 50, 54, 43, 74, 99, 76, 119, 166, 144, 182, 276, 254, 294, 442, 451, 468, 701, 758, 772, 1088, 1270, 1256, 1698, 2052, 2067, 2618, 3294, 3352, 4065, 5162, 5430, 6284, 8050
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 17 2015

Keywords

Links

Crossrefs

Cf. A262876, A263031, A263406, A263414, A263415.

Programs

  • Maple
    with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d-1, 3)=0, (d-1)/3, 0),
       d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # after Alois P. Heinz, Oct 17 2015
  • Mathematica
    nmax = 60; CoefficientList[Series[Product[1/(1-x^(3*k+1))^k,{k,1,nmax}],{x,0,nmax}],x]
nmax = 60; CoefficientList[Series[E^Sum[x^(4*k)/(k*(1-x^(3*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: exp(Sum_{k>=1} x^(4*k)/(k*(1-x^(3*k))^2)).
a(n) ~ c * Zeta(3)^(19/108) * exp(-Pi^4/(3888*Zeta(3)) - Pi^2 * n^(1/3) / (2^(4/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (2^(35/108) * 3^(23/27) * sqrt(Pi) * n^(73/108)), where c = 3^(1/6) * sqrt(2*Pi) * exp(A263031) / Gamma(1/3) = 1.107474840397395849254161220076423560365022...

A263406 Expansion of Product_{k>=1} 1/(1-x^(3*k+2))^k.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 3, 0, 2, 4, 1, 6, 5, 2, 10, 7, 6, 19, 9, 14, 29, 14, 28, 46, 23, 53, 66, 43, 95, 99, 76, 158, 143, 141, 256, 217, 247, 403, 326, 432, 617, 509, 720, 935, 801, 1187, 1399, 1281, 1892, 2087, 2047, 2983, 3107, 3272, 4589, 4647
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 17 2015

Keywords

Links

Crossrefs

Cf. A262877, A263030, A263405, A263414, A263415.

Programs

  • Maple
    with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d-2, 3)=0, (d-2)/3, 0),
       d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 17 2015
  • Mathematica
    nmax = 60; CoefficientList[Series[Product[1/(1-x^(3*k+2))^k,{k,1,nmax}],{x,0,nmax}],x]
nmax = 60; CoefficientList[Series[E^Sum[x^(5*k)/(k*(1-x^(3*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: exp(Sum_{k>=1} x^(5*k)/(k*(1-x^(3*k))^2)).
a(n) ~ c * Zeta(3)^(13/108) * exp(-Pi^4/(972*Zeta(3)) - Pi^2 * n^(1/3) / (2^(1/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (2^(41/108) * 3^(20/27) * sqrt(Pi) * n^(67/108)), where c = 3^(1/3) * Gamma(1/3) * exp(A263030) / sqrt(2*Pi) = 1.2763162741536982965216627321306598385267089489...

A263414 Expansion of Product_{k>=1} 1/(1-x^(3*k+4))^k.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 1, 0, 4, 2, 0, 5, 6, 1, 6, 10, 2, 7, 19, 6, 9, 28, 14, 11, 44, 28, 16, 61, 52, 25, 87, 93, 45, 116, 153, 77, 160, 244, 141, 215, 376, 244, 301, 560, 422, 422, 817, 695, 617, 1173, 1132, 917, 1661, 1776, 1399, 2331
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 17 2015

Keywords

Comments

In general, if v>0, GCD(v,3)=1 and g.f. = Product_{k>=1} 1/(1-x^(3*k+v))^k, then
a(n) ~ d3(v) * 3^(v^2/27 - 8/9) * exp(-Pi^4 * v^2 / (3888*Zeta(3)) - v * Pi^2 * n^(1/3) / (2^(4/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) * n^(v^2/54 - 25/36) / (sqrt(Pi) * 2^(v^2/54 + 11/36) * Zeta(3)^(v^2/54 - 7/36)), where
d3(v) = exp(Integral_{x=0..infinity} (exp((3-v)*x) / (exp(3*x)-1)^2 + (1/12 - v^2/18)/exp(x) - 1/(9*x^2) + v/(9*x))/x dx).
if mod(v,3)=1, then d3(v) = exp(A263031) * 2^((v+2)/6) * 3^((v+2)/18) * Pi^((v+2)/6) / (Gamma(1/3)^((v+2)/3) * A263416((v-1)/3)).
if mod(v,3)=2, then d3(v) = exp(A263030) * 2^((v+1)/6) * Pi^((v+1)/6) / (3^((v+1)/18) * Gamma(2/3)^((v+1)/3) * A263417((v-2)/3)).

Links

Crossrefs

Cf. A262877, A262876, A263405 (v=1), A263406 (v=2), A263415 (v=5), A263031, A263416.

Programs

  • Maple
    with(numtheory):
a:= proc(n) option remember; local r; `if`(n=0, 1,
       add(add(`if`(irem(d-3, 3, 'r')=1, d*r, 0)
       , d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Oct 17 2015
  • Mathematica
    nmax = 80; CoefficientList[Series[Product[1/(1-x^(3*k+4))^k,{k,1,nmax}],{x,0,nmax}],x]
nmax = 80; CoefficientList[Series[E^Sum[x^(7*k)/(k*(1-x^(3*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: exp(Sum_{k>=1} x^(7*k)/(k*(1-x^(3*k))^2)).
a(n) ~ c * exp(-Pi^4/(243*Zeta(3)) - 4*Pi^2 * n^(1/3) / (2^(4/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (sqrt(Pi) * 2^(65/108) * 3^(8/27) * Zeta(3)^(11/108) * n^(43/108)), where c = exp(A263031) * 2 * 3^(1/3) * Pi / Gamma(1/3)^2 = 1.24446091929106216111829684663735422946506...

A263417 a(n) = Product_{k=0..n} (3*k+2)^(n-k).

Original entry on oeis.org

1, 2, 20, 1600, 1408000, 17346560000, 3633063526400000, 15218176499384320000000, 1466155647574283911168000000000, 3672576800382377947366110003200000000000, 266783946802402043703868836144710942720000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 17 2015

Keywords

Links

Crossrefs

Cf. A263416, A263406, A263415, A369468.

Programs

  • Mathematica
    Table[Product[(3*k+2)^(n-k),{k,0,n}],{n,0,12}]
(* or *)
Table[1/FullSimplify[Gamma[2/3]^((v-2)/3) * 3^((v-2)/18) * Exp[Integrate[(E^((3-v)*x) - E^x)/(x*(E^(3*x)-1)^2) + (v-2) * (1/(3*x*(E^(3*x)-1)) + 1/(6*x*E^(3*x)) - (v+2)/(18*x*E^x)), {x, 0, Infinity}]]], {v, 2, 35, 3}]
Table[3^(n*(n+1)/2) * BarnesG[n + 5/3] / (BarnesG[2/3] * Gamma[2/3]^(n+1)), {n, 0, 12}] // Round (* Vaclav Kotesovec, Jan 23 2024 *)

Formula

a(n) ~ A^(1/3) * 2^(n/2 + 1/3) * 3^(n^2/2 + n/2 - 1/72) * Pi^(n/2 + 1/3) * n^(n^2/2 + 2*n/3 + 5/36) / (Gamma(2/3)^(n + 2/3) * exp(3*n^2/4 + 2*n/3 - Pi/(18*sqrt(3)) + PolyGamma(1, 1/3) / (12*sqrt(3)*Pi) + 1/36)), where A = A074962 is the Glaisher-Kinkelin constant and PolyGamma(1, 1/3) = 10.095597125427094081792004... (PolyGamma[1, 1/3] in Mathematica or Psi(1, 1/3) in Maple).
PolyGamma(1, 1/3) = 3^(3/2) * A261024 + 2*Pi^2/3.
a(n) = 3^(n*(n+1)/2) * BarnesG(n + 5/3) / (BarnesG(2/3) * Gamma(2/3)^(n+1)). - Vaclav Kotesovec, Jan 23 2024
Showing 1-5 of 5 results.

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