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 1-6 of 6 results.

A305699 Coefficients of 1/(q*(j(q)-744)) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, 0, -196884, -21493760, 37899009486, 8443309031424, -6829893232051144, -2454385780209696768, 1130962845597176786661, 621972524796731658731520, -164194903359722124902384028, -144508453392903668301846454272
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2018

Keywords

Links

Crossrefs

(q*(j(q)-744))^(k/4): A305698 (k=-2), A305696 (k=-1), A304020 (k=1), A305697 (k=2).
Cf. A000521 (j), A014708 (j-744), A066395, A289417.

Programs

  • Mathematica
    CoefficientList[Series[1/((2^16 + x*QPochhammer[-1, x]^24)^3/(2*QPochhammer[-1, x])^24 - 744*x), {x, 0, 15}], x] (* Vaclav Kotesovec, Jun 09 2018 *)

A302407 Euler transform is q*(j - 744) where j is j-function (A000521).

Original entry on oeis.org

0, 196884, 21493760, -18517453200, -4211531587584, 2143133970044180, 810610380990001152, -252062828441043903360, -152653440938385949943808, 26439250335625251445887252, 28061829208393772124518295552, -1741749578655115214196938355088
Offset: 1

Views

Author

Seiichi Manyama, Jun 09 2018

Keywords

Links

Crossrefs

Cf. A000521, A014708 (j-744), A192731, A304020.

Formula

q * (j(q) - 744) = Product_{k>0} (1 - x^k)^(-a(k)).

A305696 Coefficients of (q*(j(q)-744))^(-1/4) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, 0, -49221, -5373440, 5840692110, 1317368987136, -769081921703395, -285861152927176704, 99587019847435059600, 58472021328782000084480, -11456674101843809483255526, -11455351916487867258761894400, 892125673948866841204086469705
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2018

Keywords

Links

Crossrefs

(q*(j(q)-744))^(k/4): A305699 (k=-4), A305698 (k=-2), this sequence (k=-1), A304020 (k=1), A305697 (k=2).
Cf. A000521 (j), A014708 (j-744), A289397, A289416, A302407.

Programs

  • Mathematica
    CoefficientList[Series[((2^16 + x*QPochhammer[-1, x]^24)^3/(2*QPochhammer[-1, x])^24 - 744*x)^(-1/4), {x, 0, 15}], x] (* Vaclav Kotesovec, Jun 09 2018 *)

Formula

G.f.: Product_{k>0} (1 - x^k)^(A302407(k)/4).

A305697 Coefficients of (q*(j(q)-744))^(1/2) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, 0, 98442, 10746880, -4413263697, -1047821432832, 376869391313174, 150580578862513152, -35577391320709928685, -23497935558209789278208, 2998297272257446799809386, 3754973355232751413790773248, -112875007087323495790855645044
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2018

Keywords

Links

Crossrefs

(q*(j(q)-744))^(k/4): A305699 (k=-4), A305698 (k=-2), A305696 (k=-1), A304020 (k=1), this sequence (k=2).
Cf. A000521 (j), A014708 (j-744).

Programs

  • Mathematica
    CoefficientList[Series[((2^16 + x*QPochhammer[-1, x]^24)^3/(2*QPochhammer[-1, x])^24 - 744*x)^(1/2), {x, 0, 15}], x] (* Vaclav Kotesovec, Jun 09 2018 *)

A305698 Coefficients of (q*(j(q)-744))^(-1/2) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, 0, -98442, -10746880, 14104091061, 3163710154752, -2084259398665810, -764175960909112320, 294840080134539846210, 168738710694984764315648, -36893258480144387666915136, -35102639613834243676336481280
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2018

Keywords

Links

Crossrefs

(q*(j(q)-744))^(k/4): A305699 (k=-4), this sequence (k=-2), A305696 (k=-1), A304020 (k=1), A305697 (k=2).
Cf. A000521 (j), A014708 (j-744).

Programs

  • Mathematica
    CoefficientList[Series[((2^16 + x*QPochhammer[-1, x]^24)^3/(2*QPochhammer[-1, x])^24 - 744*x)^(-1/2), {x, 0, 15}], x] (* Vaclav Kotesovec, Jun 09 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.
Showing 1-6 of 6 results.

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