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 1-5 of 5 results.

A287933 Coefficients in expansion of 1/E_8.

Original entry on oeis.org

1, -480, 168480, -52199040, 15119446560, -4198347132480, 1132514464199040, -299116847254053120, 77742157641008378400, -19951615350261029163360, 5068304275307482667436480, -1276700988345016720650917760
Offset: 0

Views

Author

Seiichi Manyama, Jun 17 2017

Keywords

Links

Crossrefs

Cf. A008410 (E_8).
Cf. A288816 (k=2), A001943 (k=4), A000706 (k=6), this sequence (k=8), A285836 (k=10), A287964 (k=14).

Formula

a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) * n, where c = (262144 * Pi^24) / (81 * Gamma(1/3)^36) = 1.0839091249080051624370140889296742679583925822413671... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018

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

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

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

A289568 Coefficients in expansion of 1/E_10^(1/2).

Original entry on oeis.org

1, 132, 93852, 35163744, 18119136156, 8462089683432, 4234179302847648, 2096050696254014016, 1057219212439789539228, 534730176137991079392036, 272470142855167873443179352, 139363825115618499934478625696
Offset: 0

Views

Author

Seiichi Manyama, Jul 08 2017

Keywords

Links

Crossrefs

1/E_k^(1/2): A289565 (k=2), A289566 (k=4), A289567 (k=6), A001943 (k=8), this sequence (k=10), A289569 (k=14).
Cf. A285836 (1/E_10), A289024, A289294 (E_10^(1/2)).

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 264*Sum[DivisorSigma[9, k]*x^k, {k, 1, nmax}])^(-1/2), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 09 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(-A289024(n)/2).
a(n) ~ c * exp(2*Pi*n) / sqrt(n), where c = 0.4542595790370690606664796229968194763901027924111318430568304678613... = 2^(7/2) * Gamma(3/4)^12 / (3^(3/2) * Pi^(7/2)). - Vaclav Kotesovec, Jul 09 2017, updated Mar 07 2018
Showing 1-5 of 5 results.

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