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 1-7 of 7 results.

A288471 Exponents a(1), a(2), ... such that E_8, 1 + 480*q + 61920*q^2 + ... (A008410) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

Original entry on oeis.org

-480, 53520, -8192480, 1417877520, -261761532384, 50337746997520, -9956715872256480, 2010450258635669520, -412391756829925376480, 85648872592091236716816, -17967933476075186380800480, 3800832540589574135423637520
Offset: 1

Views

Author

Seiichi Manyama, Jun 21 2017

Keywords

Links

Crossrefs

Cf. A288968 (k=2), A110163 (k=4), A288851 (k=6), this sequence (k=8), A289024 (k=10), A288990/A288989 (k=12), A289029 (k=14).
Cf. A008410 (E_8), A008683, A288261 (E_10/E_8), A289638.

Formula

a(n) = 16 + (2/(3*n)) * Sum_{d|n} A008683(n/d) * A288261(d).
a(n) = 2 * A110163(n) = 2 * A013953(n^2). - Seiichi Manyama, Jun 22 2017
a(n) = (1/n) * Sum_{d|n} A008683(n/d) * A289638(d). - Seiichi Manyama, Jul 09 2017
a(n) ~ 2 * (-1)^n * exp(Pi*sqrt(3)*n) / n. - Vaclav Kotesovec, Mar 08 2018

A289636 Coefficients in expansion of -q*E'_4/E_4 where E_4 is the Eisenstein Series (A004009).

Original entry on oeis.org

-240, 53280, -12288960, 2835808320, -654403831200, 151013228757120, -34848505552897920, 8041801037378486400, -1855762905734676483120, 428244362959801779806400, -98823634118413525094402880, 22804995243537595828606337280
Offset: 1

Views

Author

Seiichi Manyama, Jul 09 2017

Keywords

Examples

			a(1) = 1 * A110163(1) = -240,
a(2) = 1 * A110163(1) + 2 * A110163(2) = 53280,
a(3) = 1 * A110163(1) + 3 * A110163(3) = -12288960.

Links

Crossrefs

-q*E'_k/E_k: A289635 (k=2), this sequence (k=4), A289637 (k=6), A289638 (k=8), A289639 (k=10), A289640 (k=14).
Cf. A001943, A004009 (E_4), A006352 (E_2), A110163, A145094, A289633.

Programs

  • Mathematica
    nmax = 20; Rest[CoefficientList[Series[-240*x*Sum[k*DivisorSigma[3, k]*x^(k-1), {k, 1, nmax}]/(1 + 240*Sum[DivisorSigma[3, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 09 2017 *)
terms = 12; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[-D[Ei[4], x]/Ei[4] + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)

Formula

a(n) = Sum_{d|n} d * A110163(d) = A289633(n)/6.
a(n) = A288261(n)/3 + 8*A000203(n).
a(n) = -Sum_{k=1..n-1} A004009(k)*a(n-k) - A004009(n)*n.
G.f.: 1/3 * E_6/E_4 - 1/3 * E_2.
a(n) ~ (-1)^n * exp(Pi*sqrt(3)*n). - Vaclav Kotesovec, Jul 09 2017

A289639 Coefficients in expansion of -q*E'_10/E_10 where E_10 is the Eisenstein Series (A013974).

Original entry on oeis.org

264, 340560, 141251616, 85062410400, 43377095394864, 23729517350865216, 12591243615814264896, 6769208775901467246912, 3618692733697667332476264, 1939201752717876551124987360, 1038098212042387655796115897440
Offset: 1

Views

Author

Seiichi Manyama, Jul 09 2017

Keywords

Links

Crossrefs

-q*E'_k/E_k: A289635 (k=2), A289636 (k=4), A289637 (k=6), A289638 (k=8), this sequence (k=10), A289640 (k=14).
Cf. A006352 (E_2), A013974 (E_10), A285836, A289024.

Programs

  • Mathematica
    nmax = 20; Rest[CoefficientList[Series[264*x*Sum[k*DivisorSigma[9, k]*x^(k-1), {k, 1, nmax}]/(1 - 264*Sum[DivisorSigma[9, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 09 2017 *)

Formula

a(n) = Sum_{d|n} d * A289024(d).
a(n) = A288261(n)/3 + A288840(n)/2 + 20*A000203(n).
a(n) = -Sum_{k=1..n-1} A013974(k)*a(n-k) - A013974(n)*n.
G.f.: 1/3 * E_6/E_4 + 1/2 * E_8/E_6 - 5/6 * E_2.
a(n) ~ exp(2*Pi*n). - Vaclav Kotesovec, Jul 09 2017

A289640 Coefficients in expansion of -q*E'_14/E_14 where E_14 is the Eisenstein Series (A058550).

Original entry on oeis.org

24, 393840, 128962656, 87898218720, 42722691563664, 23880530579622336, 12556395110261366976, 6777250576938845733312, 3616836970791932655993144, 1939629997080836352904793760, 1037999388408269242271021494560
Offset: 1

Views

Author

Seiichi Manyama, Jul 09 2017

Keywords

Links

Crossrefs

-q*E'_k/E_k: A289635 (k=2), A289636 (k=4), A289637 (k=6), A289638 (k=8), A289639 (k=10), this sequence (k=14).
Cf. A006352 (E_2), A058550 (E_14), A287964, A289029.

Programs

  • Mathematica
    nmax = 20; Rest[CoefficientList[Series[24*x*Sum[k*DivisorSigma[13, k]*x^(k-1), {k, 1, nmax}]/(1 - 24*Sum[DivisorSigma[13, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 09 2017 *)

Formula

a(n) = Sum_{d|n} d * A289029(d).
a(n) = 2*A288261(n)/3 + A288840(n)/2 + 28*A000203(n).
a(n) = -Sum_{k=1..n-1} A058550(k)*a(n-k) - A058550(n)*n.
G.f.: 2/3 * E_6/E_4 + 1/2 * E_8/E_6 - 7/6 * E_2.
a(n) ~ exp(2*Pi*n). - Vaclav Kotesovec, Jul 09 2017

A289635 Coefficients in expansion of -q*E'_2/E_2 where E_2 is the Eisenstein Series (A006352).

Original entry on oeis.org

24, 720, 19296, 517920, 13893264, 372707136, 9998360256, 268219317312, 7195339794744, 193024557070560, 5178140391612960, 138910500937231488, 3726458885094926160, 99967214347459657344, 2681753442755678231616
Offset: 1

Views

Author

Seiichi Manyama, Jul 09 2017

Keywords

Examples

			a(1) = - A006352(1)*1 = 24,
a(2) = -(A006352(1)*a(1)) - A006352(2)*2 = 720,
a(3) = -(A006352(1)*a(2)  + A006352(2)*a(1)) - A006352(3)*3 = 19296,
a(4) = -(A006352(1)*a(3)  + A006352(2)*a(2)  + A006352(3)*a(1)) - A006352(4)*4 = 517920.

Links

Crossrefs

-q*E'_k/E_k: this sequence (k=2), A289636 (k=4), A289637 (k=6), A289638 (k=8), A289639 (k=10), A289640 (k=14).
Cf. A000203, A006352 (E_2), A076835, A211342, A288816, A288877, A288968.

Programs

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

Formula

a(n) = Sum_{d|n} d * A288968(d).
a(n) = A288877(n)/12 + 2*A000203(n).
a(n) = -Sum_{k=1..n-1} A006352(k)*a(n-k) - A006352(n)*n.
G.f.: 1/12 * E_4/E_2 - 1/12 * E_2.
a(n) ~ 1 / r^n, where r = A211342 = 0.037276810296451658150980785651644618... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24. - Vaclav Kotesovec, Jul 09 2017

A289637 Coefficients in expansion of -q*E'_6/E_6 where E_6 is the Eisenstein Series (A013973).

Original entry on oeis.org

504, 287280, 153540576, 82226602080, 44031499226064, 23578504122108096, 12626092121367162816, 6761166974864088760512, 3620548496603402008959384, 1938773508354916749345180960, 1038197035676506069321210300320
Offset: 1

Views

Author

Seiichi Manyama, Jul 09 2017

Keywords

Links

Crossrefs

-q*E'_k/E_k: A289635 (k=2), A289636 (k=4), this sequence (k=6), A289638 (k=8), A289639 (k=10), A289640 (k=14).
Cf. A000706, A006352 (E_2), A013973 (E_6), A145095, A288851.

Programs

  • Mathematica
    nmax = 20; Rest[CoefficientList[Series[504*x*Sum[k*DivisorSigma[5, k]*x^(k-1), {k, 1, nmax}]/(1 - 504*Sum[DivisorSigma[5, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 09 2017 *)

Formula

a(n) = Sum_{d|n} d * A288851(d).
a(n) = A288840(n)/2 + 12*A000203(n).
a(n) = -Sum_{k=1..n-1} A013973(k)*a(n-k) - A013973(n)*n.
G.f.: 1/2 * E_8/E_6 - 1/2 * E_2.
a(n) ~ exp(2*Pi*n). - Vaclav Kotesovec, Jul 09 2017

A289744 Coefficients in expansion of q*E'_8 where E_8 is the Eisenstein Series (A008410).

Original entry on oeis.org

480, 123840, 3150720, 31704960, 187502400, 812885760, 2767107840, 8116473600, 20671878240, 48375619200, 102892268160, 208111357440, 391550752320, 713913822720, 1230765753600, 2077817249280, 3348363579840, 5333344585920, 8152110268800, 12384908524800
Offset: 1

Views

Author

Seiichi Manyama, Jul 11 2017

Keywords

Links

Crossrefs

(-1)^(k/2)*q*E'_{k}: A076835 (k=2), A145094 (k=4), A145095 (k=6), this sequence (k=8), A289745 (k=10), A289746 (k=14).
Cf. A008410 (E_8), A013955, A282060, A289638.

Programs

Formula

a(n) = 480*A282060(n) = 480*n*A013955(n).
Showing 1-7 of 7 results.

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