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.

A289061 a(n) = 2 * (A288851(n) - 12).

Original entry on oeis.org

984, 286752, 102360024, 41113157376, 17612599690200, 7859501322760224, 3607454891819189208, 1690291743695465539584, 804566332578533745648600, 387754701670974543569133600, 188763097395728376240220054488
Offset: 1

Views

Author

Seiichi Manyama, Jun 23 2017

Keywords

Links

Crossrefs

Related to E_{k+2}/E_k: A288995 (k=2), A192731 (k=4), this sequence (k=6).
Cf. A008683, A288840 (E_8*E_6), A288851.
Cf. A289063.

Formula

a(n) = (1/n) * Sum_{d|n} A008683(n/d) * A288840(d).
a(n) ~ 2*exp(2*Pi*n) / n. - Vaclav Kotesovec, Jun 03 2018

A289062 Coefficients in expansion of E_2^12/Product_{k>=1} (1-q^k)^24.

Original entry on oeis.org

1, -264, 30564, -2012800, 81099090, -1952940672, 22697326712, 63468624384, -4486982088465, 11373493964160, 616923039055284, -663002527580928, -77516928995402226, -352040146340083200, 5929423960701095640, 87636971447313802240, 269600086946598203619
Offset: 0

Views

Author

Seiichi Manyama, Jun 23 2017

Keywords

Examples

			G.f.: (1-q)^264 * (1-q^2)^4152 * (1-q^3)^77064 * ... = 1 - 264*q + 30564*q^2 - 2012800*q^3  + 81099090*q^4 - 1952940672*q^5 + ... .

Links

Crossrefs

Cf. A000594, A006352 (E_2), A288995, A289063.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 24*Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}])^12 / Product[(1 - x^k)^24, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 09 2017 *)

Formula

G.f.: Product_{k>=1} (1-q^k)^A288995(k).
a(n) ~ exp(4*Pi*sqrt(n)) * n^(21/4) / sqrt(2). - Vaclav Kotesovec, Jul 09 2017

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

A289350 Coefficients in expansion of E_2/Product_{k>=1} (1-q^k)^2.

Original entry on oeis.org

1, -22, -115, -350, -940, -2124, -4615, -9130, -17575, -32100, -57239, -98512, -166595, -274350, -445055, -708124, -1112002, -1719410, -2629450, -3970230, -5937238, -8785502, -12889630, -18741250, -27045445, -38724088, -55074057, -77791320, -109215025
Offset: 0

Views

Author

Seiichi Manyama, Jul 03 2017

Keywords

Crossrefs

E_2^(m/2)/Product_{k>=1} (1-q^k)^m: A289344 (m=1), this sequence (m=2), A289062 (m=24).
Cf. A006352 (E_2), A066186, A288995.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[(1 - 24*Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}]) / Product[(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Formula

G.f.: Product_{k>=1} (1-q^k)^(A288995(k)/12).
a(n) ~ -3^(1/4) * exp(2*Pi*sqrt(n/3)) / n^(1/4). - Vaclav Kotesovec, Jul 08 2017
Showing 1-4 of 4 results.

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