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.

A288851 Exponents a(1), a(2), ... such that E_6, 1 - 504*q - 16632*q^2 - ... (A013973) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

Original entry on oeis.org

504, 143388, 51180024, 20556578700, 8806299845112, 3929750661380124, 1803727445909594616, 845145871847732769804, 402283166289266872824312, 193877350835487271784566812, 94381548697864188120110027256, 46328820782943001597184984563596
Offset: 1

Views

Author

Seiichi Manyama, Jun 18 2017

Keywords

Links

Crossrefs

Cf. A288968 (k=2), A110163 (k=4), this sequence (k=6), A288471 (k=8), A289024 (k=10), A288990/A288989 (k=12), A289029 (k=14).
Cf. A008683, A013973 (E_6), A110163, A288840 (E_8/E_6), A289637.

Formula

a(n) = A013975(n^2) for n>=1.
a(n) = 12 + (1/(2*n)) * Sum_{d|n} A008683(n/d) * A288840(d).
a(n) = (1/n) * Sum_{d|n} A008683(n/d) * A289637(d). - Seiichi Manyama, Jul 09 2017
a(n) ~ exp(2*Pi*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

A289638 Coefficients in expansion of -q*E'_8/E_8 where E_8 is the Eisenstein Series (A008410).

Original entry on oeis.org

-480, 106560, -24577920, 5671616640, -1308807662400, 302026457514240, -69697011105795840, 16083602074756972800, -3711525811469352966240, 856488725919603559612800, -197647268236827050188805760, 45609990487075191657212674560
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), this sequence (k=8), A289639 (k=10), A289640 (k=14).
Cf. A006352 (E_2), A008410 (E_8), A287933, A288471.

Programs

  • Mathematica
    nmax = 20; Rest[CoefficientList[Series[-480*x*Sum[k*DivisorSigma[7, k]*x^(k-1), {k, 1, nmax}]/(1 + 480*Sum[DivisorSigma[7, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 09 2017 *)

Formula

a(n) = Sum_{d|n} d * A288471(d).
a(n) = 2*A288261(n)/3 + 16*A000203(n).
a(n) = -Sum_{k=1..n-1} A008410(k)*a(n-k) - A008410(n)*n.
G.f.: 2/3 * E_6/E_4 - 2/3 * E_2 = 2/3 * E_10/E_8 - 2/3 * E_2.
a(n) ~ 2 * (-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
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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