A294978 -id:A294978 - cp's OEIS frontend

A289369 Coefficients in expansion of (E_4^3/E_6^2)^(1/24).

1, 72, 11232, 5461344, 2029222656, 924074630640, 411487620614784, 192705317913673152, 91031590937141544960, 43814578627107100088424, 21291642032558036150652480, 10450287314646252538819378464, 5166676457072455262194208351232

Offset: 0

Seiichi Manyama, Jul 04 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..367

(E_4^3/E_6^2)^(k/288): A289365 (k=1), A299694 (k=2), A299696 (k=3), A299697 (k=4), A299698 (k=6), A299943 (k=8), A299949 (k=9), this sequence (k=12), A299950 (k=16), A299951 (k=18), A299953 (k=24), A299993 (k=32), A299994 (k=36), A300052 (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).
Cf. A108091 (E_4^(1/8)), A109817 (E_6^(1/12)).
Cf. A289367, A294978, A294979.

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

G.f.: Product_{n>=1} (1-q^n)^(-12*A289367(n)).
a(n) ~ c * exp(2*Pi*n) / n^(11/12), where c = 2^(1/3) * Pi^(1/4) / (3^(1/24) * Gamma(1/12) * Gamma(1/4)^(1/3)) =  0.0907014320494145997187363667820553893... - Vaclav Kotesovec, Jul 08 2017, updated Mar 04 2018
a(n) * A289368(n) ~ -(sqrt(3)-1) * exp(4*Pi*n) / (24*sqrt(2)*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A295788 Coefficients in expansion of (E_10/E_2^10)^(1/4).

Original entry on oeis.org

1, -6, -41652, -11504904, -4378103178, -1652544433080, -700184843900712, -302796005909941632, -136251754253507319300, -62421509259448987324542, -29147951871527035454309160, -13787807362002100397282325912

Offset: 0

Seiichi Manyama, Feb 13 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..367

Cf. A110150, A294976, A294978, A299712.

terms = 12;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E10[x_] = 1 - 264*Sum[k^9*x^k/(1 - x^k), {k, 1, terms}];
(E10[x]/E2[x]^10)^(1/4) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

a(n) ~ -Pi^4 * exp(2*Pi*n) / (3^(7/4) * 2^(15/4) * Gamma(3/4)^7 * n^(5/4)). - Vaclav Kotesovec, Jun 03 2018

A299712 Coefficients in expansion of (E_14/E_2^14)^(1/4).

Original entry on oeis.org

1, 78, -44928, -14386944, -5323508814, -1996794824544, -833028042023424, -358702721913389568, -160514702770156497360, -73334654476723097306706, -34151846554093744054455552, -16125009656471947012310740224

Offset: 0

Seiichi Manyama, Feb 17 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..367

Cf. A294976, A294978, A295788, A299713.

terms = 12;
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}];
E10[x_] = 1 - 264*Sum[k^9*x^k/(1 - x^k), {k, 1, terms}];
E14[x_] = E4[x]*E10[x];
(E14[x]/E2[x]^14)^(1/4) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

a(n) ~ -2^(3/4) * sqrt(3) * Pi^(11/2) * exp(2*Pi*n) / (864 * Gamma(3/4)^9 * n^(5/4)). - Vaclav Kotesovec, Jun 03 2018

cp's OEIS Frontend

A289369 Coefficients in expansion of (E_4^3/E_6^2)^(1/24).

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A295788 Coefficients in expansion of (E_10/E_2^10)^(1/4).

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A299712 Coefficients in expansion of (E_14/E_2^14)^(1/4).

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