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.

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

A288816 Coefficients in expansion of 1/E_2.

Original entry on oeis.org

1, 24, 648, 17376, 466152, 12505104, 335466144, 8999325120, 241418862504, 6476381979576, 173737557697968, 4660740989265312, 125030574027131424, 3354111390776151504, 89978497733627940672, 2413792838444465745216, 64753202305891291798824
Offset: 0

Views

Author

Seiichi Manyama, Jun 17 2017

Keywords

Links

Crossrefs

Cf. A006352 (E_2).
Cf. this sequence (k=2), A001943 (k=4), A000706 (k=6), A287933 (k=8), A285836 (k=10), A287964 (k=14).

Formula

G.f.: 1/(1 - 24*sum(k>=1, k*x^k/(1 - x^k))).
a(n) ~ c / r^n, where r = A211342 = 0.037276810296451658150980785651644618... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24 and c = 0.900080074462078245744608120875628441926356101483729... - Vaclav Kotesovec, Jul 02 2017

A285836 Coefficients in expansion of 1/E_10.

Original entry on oeis.org

1, 264, 205128, 95104416, 54329698632, 28308006715824, 15339873507244704, 8172566140980183360, 4385988806258507934024, 2346434028637391065282536, 1257009611855633134427201328, 672999598306502464042506285792
Offset: 0

Views

Author

Seiichi Manyama, Jun 17 2017

Keywords

Links

Crossrefs

Cf. A013974 (E_10).
Cf. A288816 (k=2), A001943 (k=4), A000706 (k=6), A287933 (k=8), this sequence (k=10), A287964 (k=14).

Formula

a(n) ~ c * exp(2*Pi*n), where c = 128 * Gamma(3/4)^24 / (27 * Pi^6) = 0.648273189440897942951926047466605067667211940159693598407336163991191821438... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018

A287964 Coefficients in expansion of 1/E_14.

Original entry on oeis.org

1, 24, 197208, 47715936, 42451725912, 18015200386704, 10924205579505504, 5511557851517150400, 3039496830486964153944, 1604976096786795234999096, 865212805864755380070382608, 461861254217266216545148291872
Offset: 0

Views

Author

Seiichi Manyama, Jun 17 2017

Keywords

Links

Crossrefs

Cf. A058550 (E_14).
Cf. A288816 (k=2), A001943 (k=4), A000706 (k=6), A287933 (k=8), A285836 (k=10), this sequence (k=14).

Programs

  • Mathematica
    terms = 12; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[1/Ei[14] + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)

Formula

a(n) ~ c * exp(2*Pi*n), where c = 512 * Gamma(3/4)^32 / (81 * Pi^8) = 0.445315094156993820198784527343140685155693441915367780875399576353998457... - Vaclav Kotesovec, Jul 02 2017, updated Mar 07 2018

A294183 Coefficients in expansion of E_6/E_8.

Original entry on oeis.org

1, -984, 393768, -129252576, 38684099112, -10970838627984, 3003345011096352, -801909012374388672, 210169391033048138280, -54295810529811041175672, 13867098270790394508774768, -3508693915623201191415922848
Offset: 0

Views

Author

Seiichi Manyama, Feb 11 2018

Keywords

Links

Crossrefs

Cf. A008410 (E_8). A013973 (E_6), A287933, A288840.
E_k/E_{k+2}: A294181 (k=2), A294182 (k=4), this sequence (k=6).

Programs

  • Mathematica
    terms = 12;
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E8[x_] = 1 + 480*Sum[k^7*x^k/(1 - x^k), {k, 1, terms}];
E6[x]/E8[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

Formula

Convolution inverse of A288840.
a(n) ~ (-1)^n * 512 * Pi^12 * exp(Pi*sqrt(3)*n) * n / (3 * Gamma(1/3)^18). - Vaclav Kotesovec, Jun 03 2018

A378468 Coefficients in expansion of (1/E_4)^3.

Original entry on oeis.org

1, -720, 339120, -132039360, 46081214640, -14974899930720, 4627836408778560, -1377759164154871680, 398508058352289409200, -112648427646194257313040, 31252327416307233967209120, -8536592939398421710286859840, 2301363255613811638678456000320, -613491781086725734777586106900960
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 27 2024

Keywords

Crossrefs

Cf. A004009, A001943, A287933, A378469.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1+240*Sum[DivisorSigma[3, k]*x^k, {k, 1, nmax}])^(-3), {x, 0, nmax}], x]

Formula

a(n) ~ (-1)^n * 67108864 * Pi^36 * n^2 * exp(Pi*sqrt(3)*n) / (729 * Gamma(1/3)^54).

A378469 Coefficients in expansion of (1/E_4)^4.

Original entry on oeis.org

1, -960, 567360, -266138880, 108735481920, -40500351480960, 14114830665358080, -4678563821426250240, 1491145606587529742400, -460511820740945555286720, 138585483759128030100927360, -40812342463218781348220286720, 11800049457060387849887324117760, -3358272262154871467174772417214080
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 27 2024

Keywords

Comments

In general, for k > 0, the expansion of 1/(E_4)^k is asymptotic to (-1)^n * k * 2^(9*k) * Pi^(12*k) * n^(k-1) * exp(Pi*sqrt(3)*n) / (3^(2*k) * Gamma(1/3)^(18*k) * Gamma(k+1)).

Crossrefs

Cf. A001943 (k=1), A287933 (k=2), A378468 (k=3).
Cf. A289566 (k=1/2), A295815 (k=1/4), A289247 (k=1/8).
Cf. A004009, A289319.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1+240*Sum[DivisorSigma[3, k]*x^k, {k, 1, nmax}])^(-4), {x, 0, nmax}], x]

Formula

a(n) ~ (-1)^n * 34359738368 * Pi^48 * n^3 * exp(Pi*sqrt(3)*n) / (19683 * Gamma(1/3)^72).
Showing 1-7 of 7 results.

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