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.

A288968 Exponents a(1), a(2), ... such that E_2, 1 - 24*q - 72*q^2 - ... (A006352) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

Original entry on oeis.org

24, 348, 6424, 129300, 2778648, 62114524, 1428337176, 33527349924, 799482197272, 19302454317660, 470740035601176, 11575875047000596, 286650683468840472, 7140515309818664028, 178783562850377621272, 4496350112540599930692
Offset: 1

Views

Author

Seiichi Manyama, Jun 20 2017

Keywords

Links

Crossrefs

Cf. this sequence (k=2), A110163 (k=4), A288851 (k=6), A288471 (k=8), A289024 (k=10), A288990/A288989 (k=12), A289029 (k=14).
Cf. A006352 (E_2), A008683, A288877 (E_4/E_2), A289635.

Formula

a(n) = 2 + (1/(12*n)) * Sum_{d|n} A008683(n/d) * A288877(d).
a(n) = (1/n) * Sum_{d|n} A008683(n/d) * A289635(d).
a(n) ~ 1 / (n * r^(2*n)), where r = A057823. - 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

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
Showing 1-6 of 6 results.

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