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.

A289368 Coefficients in expansion of (E_6^2/E_4^3)^(1/24).

Original entry on oeis.org

1, -72, -6048, -4217184, -1264437504, -606533479920, -251777443450752, -117085712395216320, -53634689421870422016, -25408429618361083967592, -12110787335129301116994240, -5854620911089647830793873696
Offset: 0

Views

Author

Seiichi Manyama, Jul 04 2017

Keywords

Links

Crossrefs

(E_6^2/E_4^3)^(k/288): A289366 (k=1), A296609 (k=2), A296614 (k=3), A296652 (k=4), A297021 (k=6), A299422 (k=8), A299862 (k=9), this sequence (k=12), A299856 (k=16), A299857 (k=18), A299858 (k=24), A299863 (k=32), A299859 (k=36), A299860 (k=48), A299861 (k=72), A299414 (k=96), A299413 (k=144), A289210 (k=288).
Cf. A000521 (j), A108091 (E_4^(1/8)), A109817 (E_6^(1/12)).
Cf. A289366, A289367, A294974, A294976.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[((1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^2 / (1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^3)^(1/24), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Formula

G.f.: (1 - 1728/j)^(1/24).
G.f.: Product_{n>=1} (1-q^n)^(12*A289367(n)).
a(n) ~ c * exp(2*Pi*n) / n^(13/12), where c = -Gamma(1/4)^(1/3) / (2^(7/3) * 3^(23/24) * Pi^(1/4) * Gamma(11/12)) = -0.07569217204117312767729284017524325060022536591050774997610261275428... - Vaclav Kotesovec, Jul 08 2017, updated Mar 04 2018
a(n) * A289369(n) ~ -(sqrt(3)-1) * exp(4*Pi*n) / (24*sqrt(2)*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A294976 Coefficients in expansion of (E_6/E_2^6)^(1/12).

Original entry on oeis.org

1, -30, -11340, -3912600, -1520905170, -636170644008, -278687199310200, -126000360658968000, -58290111778749466140, -27440829122946510954630, -13096614404248661886145848, -6320198941502349713305002120, -3077986352751848627729986859400
Offset: 0

Views

Author

Seiichi Manyama, Feb 12 2018

Keywords

Crossrefs

Cf. A109817, A289565, A294974, A294975.

Programs

  • Mathematica
    terms = 13;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
(E6[x]/E2[x]^6)^(1/12) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^A294975(n).
a(n) ~ -Gamma(1/3)^2 * Gamma(1/4)^(10/3) * exp(2*Pi*n) / (16 * 2^(1/12) * 3^(7/12) * Pi^(5/2) * Gamma(1/12) * n^(13/12)). - Vaclav Kotesovec, Jun 03 2018
Equivalently, a(n) ~ -Gamma(1/3) * Gamma(1/4)^(7/3) * exp(2*Pi*n) / (2^(23/6) * 3^(23/24) * Pi^2 * sqrt(1 + sqrt(3)) * n^(13/12)). - Vaclav Kotesovec, Nov 26 2024

A294626 a(n) = (1/(24*n)) * Sum_{d|n} A008683(n/d) * (A288877(d) - A288261(d)).

Original entry on oeis.org

42, -3171, 515242, -88552695, 16361485098, -3146078130083, 622295456184618, -125653124401054383, 25774485201611434666, -5353054527354475135971, 1122995842490069166600618, -237552033781060445940477047, 50601782105864798623718932266
Offset: 1

Views

Author

Seiichi Manyama, Feb 12 2018

Keywords

Links

Crossrefs

Cf. A008683, A288261, A288877, A294974.

Programs

  • Mathematica
    terms = 13;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
a[n_] := (1/(24 n))*Sum[MoebiusMu[n/d]*SeriesCoefficient[E4[x]/E2[x] - E6[x]/E4[x], {x, 0, d}], {d, Divisors[n]}];
Array[a, terms] (* Jean-François Alcover, Feb 26 2018 *)

Formula

a(n) ~ -(-1)^n * exp(Pi*sqrt(3)*n) / (8*n). - Vaclav Kotesovec, Jun 03 2018

A294978 Coefficients in expansion of (E_4/E_2^4)^(1/8).

Original entry on oeis.org

1, 42, -2268, 395304, -64600914, 11644170552, -2188350306072, 424652412357696, -84326944950450972, 17044476557469661986, -3493525880987663047128, 724189608821718233434296, -151528575864988356484968840, 31955212589107172812017247992
Offset: 0

Views

Author

Seiichi Manyama, Feb 12 2018

Keywords

Comments

Also coefficients in expansion of (E_8/E_2^8)^(1/16).

Crossrefs

Cf. A004009 (E_4), A006352 (E_2), A108091, A289565, A294626, A294974.

Programs

  • Mathematica
    terms = 14;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
(E4[x]/E2[x]^4)^(1/8) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Formula

Convolution inverse of A294974.
G.f.: Product_{n>=1} (1-q^n)^(-A294626(n)).
a(n) ~ -(-1)^n * Pi^(5/4) * exp(Pi*sqrt(3)*n) / (2^(19/8) * 3^(9/8) * Gamma(2/3)^(9/4) * Gamma(7/8) * n^(9/8)). - Vaclav Kotesovec, Jun 03 2018

A299713 Coefficients in expansion of (E_2^14/E_14)^(1/4).

Original entry on oeis.org

1, -78, 51012, 6903624, 5954711646, 2161166435064, 1147125232110312, 525629755914843840, 262146899489557893876, 127930551245366132625258, 63840206028392497556267688, 31874918557295958327688144536, 16048808078655467936667484701336
Offset: 0

Views

Author

Seiichi Manyama, Feb 17 2018

Keywords

Links

Crossrefs

Cf. A294974, A294979, A295789, A295817, A299712.

Formula

Convolution inverse of A299712.
a(n) ~ 2^(7/4) * 3^(5/2) * Gamma(3/4)^9 * exp(2*Pi*n) / (Pi^(13/2) * n^(3/4)). - Vaclav Kotesovec, Jun 03 2018
Showing 1-5 of 5 results.

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