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.

A288877 Coefficients in expansion of E_4/E_2.

Original entry on oeis.org

1, 264, 8568, 231456, 6214872, 166719024, 4472485344, 119980322880, 3218631807384, 86344077536616, 2316294684846288, 62137684699355232, 1666926011246777184, 44717506621139113584, 1199606572169515887552, 32181041313068138778816
Offset: 0

Views

Author

Seiichi Manyama, Jun 18 2017

Keywords

Links

Crossrefs

E_{k+2}/E_k: this sequence (k=2), A288261 (k=4), A288840 (k=6).
Cf. A004009 (E_4), A006352 (E_2), A288816 (1/E_2).
Cf. A211342.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3, k]*x^k, {k, 1, nmax}])/(1 - 24*Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 28 2017 *)
terms = 16; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[Ei[4]/Ei[2] + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)

Formula

a(n) ~ 12 / r^n, where r = A211342 = 0.037276810296451658150980785651644618... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24. - Vaclav Kotesovec, Jun 28 2017

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

A289565 Coefficients in expansion of 1/E_2^(1/2).

Original entry on oeis.org

1, 12, 252, 5664, 133356, 3224952, 79387488, 1978996416, 49797787788, 1262193008556, 32177428972632, 824182154521056, 21193138994244960, 546767126418119352, 14146104826919725632, 366887630982365262144, 9535791498480146879436
Offset: 0

Views

Author

Seiichi Manyama, Jul 08 2017

Keywords

Links

Crossrefs

1/E_k^(1/2): this sequence (k=2), A289566 (k=4), A289567 (k=6), A001943 (k=8), A289568 (k=10), A289569 (k=14).
Cf. A288816 (1/E_2), A288968, A289291 (E_2^(1/2)).

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 24*Sum[DivisorSigma[1, 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)^(-A288968(n)/2).
a(n) ~ c / (sqrt(n) * r^n), where r = A211342 = 0.03727681029645165815098078565... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24 and c = 0.535261044779387956394739769118415667289349331646288208543596374426... - 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

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
Showing 1-6 of 6 results.

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