A013973 -id:A013973 - cp's OEIS frontend

A299953 Coefficients in expansion of (E_4^3/E_6^2)^(1/12).

1, 144, 27648, 12540096, 4971036672, 2263040955360, 1031452724072448, 487587831652591488, 233267529030162186240, 113311495859272029716688, 55566291037565862262794240, 27487705978359515260636550208, 13689979692617556597746930024448

Offset: 0

Seiichi Manyama, Feb 22 2018

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), A289369 (k=12), A299950 (k=16), A299951 (k=18), this sequence (k=24), A299993 (k=32), A299994 (k=36), A300052 (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).

Cf. A004009 (E_4), A013973 (E_6), A299858.

terms = 13;
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}];
(E4[x]^3/E6[x]^2)^(1/12) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Convolution inverse of A299858.
a(n) ~ 2^(2/3) * sqrt(Pi) * exp(2*Pi*n) / (3^(1/12) * Gamma(1/6) * Gamma(1/4)^(2/3) * n^(5/6)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A299858(n) ~ -exp(4*Pi*n) / (12*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299993 Coefficients in expansion of (E_4^3/E_6^2)^(1/9).

Original entry on oeis.org

1, 192, 41472, 18342144, 7524397056, 3440911653504, 1589472997005312, 756816895536990720, 364982499184388898816, 178417371665487543380928, 88017286719942539086814208, 43770603489875525093472688896, 21905830503405563891572154843136

Offset: 0

Seiichi Manyama, Feb 22 2018

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), A289369 (k=12), A299950 (k=16), A299951 (k=18), A299953 (k=24), this sequence (k=32), A299994 (k=36), A300052 (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).

Cf. A004009 (E_4), A013973 (E_6), A299863.

terms = 13;
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}];
(E4[x]^3/E6[x]^2)^(1/9) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Convolution inverse of A299863.
a(n) ~ 2^(8/9) * Pi^(2/3) * exp(2*Pi*n) / (3^(1/9) * Gamma(2/9) * Gamma(1/4)^(8/9) * n^(7/9)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A299863(n) ~ -2*sin(2*Pi/9) * exp(4*Pi*n) / (9*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299994 Coefficients in expansion of (E_4^3/E_6^2)^(1/8).

Original entry on oeis.org

1, 216, 49248, 21609504, 9000122112, 4129083886032, 1919370450227328, 917374442680570176, 444151666318727522304, 217813424092164713883960, 107771776495186976967396672, 53736084111333058216805911392, 26958647064591216695092188902400

Offset: 0

Seiichi Manyama, Feb 22 2018

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), A289369 (k=12), A299950 (k=16), A299951 (k=18), A299953 (k=24), A299993 (k=32), this sequence (k=36), A300052 (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).

Cf. A004009 (E_4), A013973 (E_6), A299859.

terms = 13;
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}];
(E4[x]^3/E6[x]^2)^(1/8) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Convolution inverse of A299859.
a(n) ~ 2 * Pi^(3/4) * exp(2*Pi*n) / (3^(1/8) * Gamma(1/4)^2 * n^(3/4)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A299859(n) ~ -exp(4*Pi*n) / (4*sqrt(2)*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A300052 Coefficients in expansion of (E_4^3/E_6^2)^(1/6).

Original entry on oeis.org

1, 288, 76032, 33042816, 14318032896, 6651157620672, 3146793694792704, 1522045714678435584, 745464270665241870336, 369134048335617435664800, 184269983601798163049283072, 92610644166133510115124717696

Offset: 0

Seiichi Manyama, Feb 23 2018

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), A289369 (k=12), A299950 (k=16), A299951 (k=18), A299953 (k=24), A299993 (k=32), A299994 (k=36), this sequence (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).

Cf. A004009 (E_4), A013973 (E_6), A299860.

terms = 12;
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}];
(E4[x]^3/E6[x]^2)^(1/6) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Convolution inverse of A299860.
a(n) ~ 2^(4/3) * Pi * exp(2*Pi*n) / (3^(1/6) * Gamma(1/4)^(4/3) * Gamma(1/3) * n^(2/3)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A299860(n) ~ -exp(4*Pi*n) / (2*sqrt(3)*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A300053 Coefficients in expansion of (E_4^3/E_6^2)^(1/4).

Original entry on oeis.org

1, 432, 145152, 64494144, 29760915456, 14274670230432, 6975951829890048, 3459591515857458816, 1733116511275051696128, 875135886353582630388336, 444632598699435462934282752, 227042568315636603738176892096

Offset: 0

Seiichi Manyama, Feb 23 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..366

(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), A289369 (k=12), A299950 (k=16), A299951 (k=18), A299953 (k=24), A299993 (k=32), A299994 (k=36), A300052 (k=48), this sequence (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).

Cf. A004009 (E_4), A013973 (E_6), A299861.

terms = 12;
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}];
(E4[x]^3/E6[x]^2)^(1/4) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Convolution inverse of A299861.
a(n) ~ 4 * Pi * exp(2*Pi*n) / (3^(1/4) * Gamma(1/4)^2 * sqrt(n)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A299861(n) ~ -exp(4*Pi*n) / (2*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A300054 Coefficients in expansion of (E_4^3/E_6^2)^(1/3).

Original entry on oeis.org

1, 576, 235008, 109880064, 53449592832, 26574124961664, 13393739222599680, 6814262482916285952, 3490692930294883909632, 1797524713443792341369664, 929454499859725260939506688, 482202319224911188610453541120

Offset: 0

Seiichi Manyama, Feb 23 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..366

(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), A289369 (k=12), A299950 (k=16), A299951 (k=18), A299953 (k=24), A299993 (k=32), A299994 (k=36), A300052 (k=48), A300053 (k=72), this sequence (k=96), A300055 (k=144), A289209 (k=288).

Cf. A004009 (E_4), A013973 (E_6), A299414.

terms = 12;
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}];
(E4[x]^3/E6[x]^2)^(1/3) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Convolution inverse of A299414.
a(n) ~ 2^(8/3) * Pi^2 * exp(2*Pi*n) / (3^(1/3) * Gamma(1/4)^(8/3) * Gamma(2/3) * n^(1/3)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A299414(n) ~ -exp(4*Pi*n) / (sqrt(3)*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A029829 Eisenstein series E_16(q) (alternate convention E_8(q)), multiplied by 3617.

Original entry on oeis.org

3617, 16320, 534790080, 234174178560, 17524001357760, 498046875016320, 7673653657232640, 77480203842286080, 574226476491096000, 3360143509958850240, 16320498047409790080, 68172690124863440640

Offset: 0

N. J. A. Sloane

nonn

N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 111.
J.-P. Serre, Course in Arithmetic, Chap. VII, Section 4.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for sequences related to Eisenstein series

Cf. A058552.

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A008410 (E_8), A013974 (E_10), A029828 (E_12), A058550 (E_14), A029829 (E_16), A029830 (E_20), A029831 (E_24).

E := proc(k) local n,t1; t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n,n=1..60); series(t1,q,60); end; E(16);

terms = 12;
E16[x_] = 3617 + 16320*Sum[k^15*x^k/(1 - x^k), {k, 1, terms}];
E16[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

a(n)=if(n<1,3617*(n==0),16320*sigma(n,15))

a(n) = 1617*A282012(n) + 2000*A282287(n). - Seiichi Manyama, Feb 11 2017

A029831 Eisenstein series E_24(q) (alternate convention E_12(q)), multiplied by 236364091.

Original entry on oeis.org

236364091, 131040, 1099243323360, 12336522153621120, 9221121336284413920, 1562118530273437631040, 103486260766565509822080, 3586400651444203277717760, 77352372210526124884754400, 1161399411211600265764157280

Offset: 0

N. J. A. Sloane

J.-P. Serre, Course in Arithmetic, Chap. VII, Section 4.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for sequences related to Eisenstein series

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A008410 (E_8), A013974 (E_10), A029828 (691*E_12), A058550 (E_14), A029829 (3617*E_16), A279892 (43867*E_18), A029830 (174611*E_20), A279893 (77683*E_22), this sequence (236364091*E_24).

Cf. A282330 (E_4^6), A282332 (E_4^3*E_6^2), A282331 (E_6^4).

terms = 10;
E24[x_] = 236364091 + 131040*Sum[k^23*x^k/(1 - x^k), {k, 1, terms}];
E24[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

a(n)=if(n<1,236364091*(n==0),131040*sigma(n,23))

a(n) = 49679091*A282330(n) + 176400000*A282332(n) + 10285000*A282331(n). - Seiichi Manyama, Feb 12 2017

A280869 Expansion of E_6(q)^2 in powers of q.

Original entry on oeis.org

1, -1008, 220752, 16519104, 399517776, 4624512480, 34423752384, 187506813312, 814794618960, 2975666040144, 9486668147040, 27052407031104, 70486610910912, 169931677686624, 384163181281152, 820165393918080, 1668889095288912, 3249638073414432

Offset: 0

Seiichi Manyama, Jan 28 2017

sign

			G.f. = 1 - 1008*q + 220752*q^2 + 16519104*q^3 + 399517776*q^4 + 4624512480*q^5 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Eisenstein Series.

Cf. A000594, A001160, A008411, A013973 (E_6), A029828 (691*E_12).

terms = 18;
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E6[x]^2 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

E6(q)^2 = (1 - 504 Sum_{i>=1} sigma_5(i)q^i)^2 where sigma_5(n) is A001160.
A008411(n) - a(n) = 1728*A000594(n).
A029828(n) - 691*a(n) = 762048*A000594(n).
A029828(n) = 441*A008411(n) + 250*a(n).

A289293 Coefficients in expansion of E_6^(1/2).

Original entry on oeis.org

1, -252, -40068, -10158624, -3362961924, -1254502939032, -502480721822688, -211053631376919744, -91717692784641665028, -40892713821496126310364, -18600635229558474625901928, -8597703758971125751979122656

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

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

E_k^(1/2): A289291 (k=2), A289292 (k=4), this sequence (k=6), A004009 (k=8), A289294 (k=10), A289295 (k=14).

Cf. A013973 (E_6), A288851.

terms = 12;
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E6[x]^(1/2) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

G.f.: Product_{n>=1} (1-q^n)^(A288851(n)/2).

a(n) ~ c * exp(2*Pi*n) / n^(3/2), where c = -3*sqrt(2)*Pi^(3/2) / (16*Gamma(3/4)^8) = -0.2903826839827320330247215149377503818798115... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018

cp's OEIS Frontend

A299953 Coefficients in expansion of (E_4^3/E_6^2)^(1/12).

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A299993 Coefficients in expansion of (E_4^3/E_6^2)^(1/9).

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A299994 Coefficients in expansion of (E_4^3/E_6^2)^(1/8).

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A300052 Coefficients in expansion of (E_4^3/E_6^2)^(1/6).

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A300053 Coefficients in expansion of (E_4^3/E_6^2)^(1/4).

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A300054 Coefficients in expansion of (E_4^3/E_6^2)^(1/3).

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A029829 Eisenstein series E_16(q) (alternate convention E_8(q)), multiplied by 3617.

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A029831 Eisenstein series E_24(q) (alternate convention E_12(q)), multiplied by 236364091.

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A280869 Expansion of E_6(q)^2 in powers of q.

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