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.

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A289562 Coefficients of 1/(q*(j(q)-1728))^3 where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, 2952, 5218884, 7138351488, 8319960432666, 8678332561127616, 8338315178481134040, 7518590274496806176256, 6444205834302869333758299, 5298802621872639665867604832, 4208666443076672300677008045636, 3246069554930472099322915758511872
Offset: 0

Views

Author

Seiichi Manyama, Jul 08 2017

Keywords

Links

Crossrefs

(q*(j(q)-1728))^(k/24): A289563 (k=-96), this sequence (k=-72), A289561 (k=-48), A289417 (k=-24), A289416 (k=-1), A106203 (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), A007242 (k=12), A289063 (k=24).
Cf. A289061.

Programs

  • Mathematica
    CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(-3), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 07 2018 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(-3*A289061(n)).
a(n) ~ c * exp(2*Pi*n) * n^5, where c = Gamma(3/4)^24 * exp(6*Pi) / (4081466880 * Pi^6) = 0.0051446247390864841578336638645072392120317488530740050289688... - Vaclav Kotesovec, Mar 07 2018

A289563 Coefficients of 1/(q*(j(q)-1728))^4 where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, 3936, 8895024, 15094625920, 21336320693400, 26506772152211520, 29887990556174431424, 31237788209244729015552, 30709242534935581933885740, 28700724444538653431660487520, 25706227251014342788669659769056, 22202613798662970539127791744222592
Offset: 0

Views

Author

Seiichi Manyama, Jul 08 2017

Keywords

Links

Crossrefs

(q*(j(q)-1728))^(k/24): this sequence (k=-96), A289562 (k=-72), A289561 (k=-48), A289417 (k=-24), A289416 (k=-1), A106203 (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), A007242 (k=12), A289063 (k=24).
Cf. A289061.

Programs

  • Mathematica
    CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(-4), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 07 2018 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(-4*A289061(n)).
a(n) ~ c * exp(2*Pi*n) * n^7, where c = Gamma(3/4)^32 * exp(8*Pi) / (55540601303040 * Pi^8) = 0.0001042996202910562374208781457852661312263780276025385904... - Vaclav Kotesovec, Mar 07 2018

A305756 Coefficients of (q*(j(q)-720))^(1/24) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, 1, 8192, 707073, -754075135, -132208502783, 90102565204481, 25124693308972545, -11606164284986636798, -4751761734938773786110, 1495856955988144882193922, 890018844816101689979518466, -181104153998957724140261556733
Offset: 0

Views

Author

Seiichi Manyama, Jun 10 2018

Keywords

Comments

(Conjecture)
Let |b| = 2^p * 3^q * 5^r * ... .
And f(0) = 24, f(b) = 2^(max(0, min(3, p - 1))) * 3^(max(0, min(1, q - 1))) for |b|>0. (See A305762)
Coefficients of (q*(j(q)+b))^(1/f(b)) are integers.
Especially, coefficients of (q*(j(q)+144*k))^(1/24) are integers.
In case of b = -744, |b| = 2^3 * 3^1 * 31 and f(b) = 4. So coefficients of (q*(j(q)-744))^(1/4) are integers. (See A304020)

Links

Crossrefs

(q*(j(q)+144*k))^(1/24): A106205 (k=0), this sequence (k=-5), A106203 (k=-12).
(q*(j(q)-720))^(m/24): A305760 (m=-24), A305758 (m=-1), this sequence (m=1).
Cf. A000521, A007240 (j(q)-720), A304020, A305757, A305762.

A289340 Coefficients of (q*(j(q)-1728))^(1/3) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, -328, -41956, -8596032, -2597408634, -916285828640, -352170121921992, -143129703441671168, -60517599938503137519, -26355020095077489965264, -11743692598044815023990588, -5329748160859504303225598464
Offset: 0

Views

Author

Seiichi Manyama, Jul 02 2017

Keywords

Comments

In general, the expansion of (q*(j(q)-1728))^m, where j(q) is the elliptic modular invariant (A000521), and m <> 0, is asymptotic to exp(4*Pi*sqrt(m*n)) * m^(1/4) / (sqrt(2) * n^(3/4)) if 2*m is the positive integer, else is asymptotic to 2^(2*m) * 3^(4*m) * Pi^(2*m) * exp(2*Pi*(n-m)) / (Gamma(-2*m) * Gamma(3/4)^(8*m) * n^(2*m + 1)). - Vaclav Kotesovec, Mar 07 2018

Crossrefs

(q*(j(q)-1728))^(k/24): A106203 (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), A289339 (k=7), this sequence (k=8), A007242 (k=12), A289063 (k=24).
Cf. A289061.

Programs

  • Mathematica
    CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(1/3), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 07 2018 *)

Formula

G.f.: Product_{k>=1} (1-q^k)^(A289061(k)/3).
a(n) ~ c * exp(2*Pi*n) / n^(5/3), where c = -2^(2/3) * 3^(5/6) * exp(-2*Pi/3) * Gamma(2/3) / (Pi^(1/3) * Gamma(3/4)^(8/3)) = -0.262554753987597280323546158564... - Vaclav Kotesovec, Mar 07 2018

A289339 Coefficients of (q*(j(q)-1728))^(7/24) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, -287, -42595, -9750370, -3081185660, -1117168154431, -438204467218406, -181018051263504195, -77584080248087108885, -34183723168674046275385, -15388633770558568711781905, -7047808475666778827478858184
Offset: 0

Views

Author

Seiichi Manyama, Jul 02 2017

Keywords

Crossrefs

(q*(j(q)-1728))^(k/24): A106203 (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), this sequence (k=7), A289340 (k=8), A007242 (k=12), A289063 (k=24).
Cf. A289061.

Programs

  • Mathematica
    CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(7/24), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 07 2018 *)

Formula

G.f.: Product_{k>=1} (1-q^k)^(7*A289061(k)/24).
a(n) ~ c * exp(2*Pi*n) / n^(19/12), where c = -7 * exp(-7*Pi/12) * Gamma(1/12) / (2^(35/12) * 3^(1/12) * Pi^(17/12) * Gamma(3/4)^(1/3)) = -0.287342744567300675294730727139553541489784437990631575713791583301655... - Vaclav Kotesovec, Mar 07 2018

A289344 Coefficients in expansion of E_2^(1/2)/Product_{k>=1} (1-q^k).

Original entry on oeis.org

1, -11, -118, -1473, -23635, -434861, -8659573, -181387821, -3936961298, -87743843970, -1996149058302, -46163368994680, -1082012001849499, -25646334881233711, -613664275728573585, -14803437882920457712, -359626550280367615329
Offset: 0

Views

Author

Seiichi Manyama, Jul 03 2017

Keywords

Crossrefs

Cf. A006352 (E_2), A288995, A289062, A289291.
Cf. A106205, A106203.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Sqrt[1 - 24*Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}]] / Product[1 - x^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 03 2017 *)

Formula

G.f.: Product_{k>=1} (1-q^k)^(A288995(k)/24).
a(n) ~ c / (n^(3/2) * r^n), where r = A211342 = 0.03727681029645165815098078565... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24 and c = -0.309300289625571303778321676728514880378401177270067457514896529... - Vaclav Kotesovec, Jul 03 2017
Previous Showing 11-16 of 16 results.

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