A029828 -id:A029828 - cp's OEIS frontend

A029830 Eisenstein series E_20(q) (alternate convention E_10(q)), multiplied by 174611.

174611, 13200, 6920614800, 15341851377600, 3628395292275600, 251770019531263200, 8043563916910526400, 150465416446925500800, 1902324110996589786000, 17831242688625346952400, 132000251770026451864800, 807299993919072011054400, 4217144038884527916580800, 19297347832955888660949600

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), this sequence (174611*E_20), A279893 (77683*E_22), A029831 (236364091*E_24).

Cf. A282015 (E_4^5), A282292 (E_4^2*E_6^2 = E_10^2).

terms = 14;
E20[x_] = 174611 + 13200*Sum[k^19*x^k/(1 - x^k), {k, 1, terms}];
E20[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

a(n)=if(n<1,174611*(n==0),13200*sigma(n,19))

a(n) = 53361*A282015(n) + 121250*A282292(n). - Seiichi Manyama, Feb 11 2017

A288989 Denominators of coefficients in expansion of E_14/E_12.

Original entry on oeis.org

1, 691, 477481, 329939371, 227988105361, 157539780804451, 108859988535875641, 75222252078290067931, 51978576186098436940321, 35917196144594019925761811, 24818782535914467768701411401, 17149778732316897228172675278091

Offset: 0

Seiichi Manyama, Jun 21 2017

			E_14/E_12 = 1 - 82104/691 * q - 181275671592/477481 * q^2  + 1327007921039904/329939371 * q^3 + 16726528971891002133912/227988105361 * q^4 + ... .

Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A288472 (numerators).

Cf. A029828, A058550 (E_14).

terms = 12;
E14[x_] = 1 - 24*Sum[k^13*x^k/(1 - x^k), {k, 1, terms}];
E12[x_] = 1 + (65520/691)*Sum[k^11*x^k/(1 - x^k), {k, 1, terms}];
E14[x]/E12[x] + O[x]^terms // CoefficientList[#, x]& // Denominator (* Jean-François Alcover, Feb 26 2018 *)

A279892 Eisenstein series E_18(q) (alternate convention E_9(q)), multiplied by 43867.

Original entry on oeis.org

43867, -28728, -3765465144, -3709938631392, -493547047383096, -21917724609403728, -486272786232443616, -6683009405824511424, -64690198594597187640, -479102079577959825624, -2872821917728374840144, -14520482234727711482016, -63736746640768788267744

Offset: 0

Seiichi Manyama, Dec 22 2016

sign

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

Seiichi Manyama, Table of n, a(n) for n = 0..10000

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), this sequence (43867*E_18), A029830 (174611*E_20), A279893 (77683*E_22), A029831 (236364091*E_24).

Cf. A282000 (E_4^3*E_6), A282253 (E_6^3).

terms = 13;
E18[x_] = 43867 - 28728*Sum[k^17*x^k/(1 - x^k), {k, 1, terms}];
E18[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

G.f.: 43867 - 28728 * Sum_{i>=1} sigma_17(i)q^i where sigma_17(n) is A013965.

a(n) = 38367*A282000(n) + 5500*A282253(n). - Seiichi Manyama, Feb 11 2017

A279893 Eisenstein series E_22(q) (alternate convention E_11(q)), multiplied by 77683.

Original entry on oeis.org

77683, -552, -1157628456, -5774114968608, -2427722831757864, -263214111328125552, -12109202528761173024, -308317316973972772416, -5091303792066668003880, -60399282006368937251976, -552000263214112485753456, -4084937969230504375869024, -25394838301602325644596256, -136379620048544616772836528, -646588586243917921590531648

Offset: 0

Seiichi Manyama, Dec 22 2016

sign

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

Seiichi Manyama, Table of n, a(n) for n = 0..10000

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), this sequence (77683*E_22), A029831 (236364091*E_24).

Cf. A282047 (E_4^4*E_6), A282328 (E_4*E_6^3).

terms = 15;
E22[x_] = 77683 - 552*Sum[k^21*x^k/(1 - x^k), {k, 1, terms}];
E22[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

G.f.: 77683 - 552 * Sum_{i>=1} sigma_21(i)q^i where sigma_21(n) is A013969.

a(n) = 57183*A282047(n) + 20500*A282328(n). - Seiichi Manyama, Feb 12 2017

A282356 Eisenstein series E_26(q) (alternate convention E_13(q)), multiplied by 657931.

Original entry on oeis.org

657931, -24, -805306392, -20334926626656, -27021598569529368, -7152557373046875024, -682326933054044766048, -32185646871935157619392, -906694391732570450559000, -17229551704624797057112632, -240000007152557373852181392

Offset: 0

Seiichi Manyama, Feb 13 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000
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), A029831 (236364091*E_24), this sequence (657931*E_26).

Cf. A282048 (E_4^5*E_6), A282357 (E_4^2*E_6^3).

terms = 11;
E26[x_] = 657931 - 24*Sum[k^25*x^k/(1 - x^k), {k, 1, terms}];
E26[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

a(n) = 392931*A282048(n) + 265000*A282357(n).

A056947 Theta series of nonexistent Niemeier lattice of Coxeter number 1.

Original entry on oeis.org

1, 24, 195984, 16779168, 397998672, 4629497040, 34417510848, 187489533504, 814881802320, 2975548760568, 9486548517600, 27052958750688, 70486228096704, 169931081461008, 384163595996544, 820166650027200, 1668890114013264, 3249630946490544, 6096882726702288

Offset: 0

Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jul 17 2000

nonn

			G.f.: 1 + 24*q + 195984*q^2 + 16779168*q^3 + ...

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, 1988.

Andy Huchala, Table of n, a(n) for n = 0..20000

Cf. A029828, A000594.

d = CuspForms(1, 12).0.q_expansion(20);
e =(eisenstein_series_qexp(12,20, normalization='integral'))
list(e/691-(48936/691)*d) # Andy Huchala, Jul 10 2021

E_12(z)+(24h-c12)*D(12) where D(12) is unique cusp form of weight 12, c12=(2*Pi)^12/(zeta(12)*gamma(12)) and h=1.

a(n) = (A029828(n) - 48936*A000594(n))/691. - Andy Huchala, Jul 11 2021

A145154 Coefficients in expansion of Eisenstein series E_1.

Original entry on oeis.org

1, 4, 8, 8, 12, 8, 16, 8, 16, 12, 16, 8, 24, 8, 16, 16, 20, 8, 24, 8, 24, 16, 16, 8, 32, 12, 16, 16, 24, 8, 32, 8, 24, 16, 16, 16, 36, 8, 16, 16, 32, 8, 32, 8, 24, 24, 16, 8, 40, 12, 24, 16, 24, 8, 32, 16, 32, 16, 16, 8, 48

Offset: 0

N. J. A. Sloane, Feb 28 2009

nonn

			1 + 4*q + 8*q^2 + 8*q^3 + 12*q^4 + 8*q^5 + 16*q^6 + 8*q^7 + 16*q^8 + ...

Antti Karttunen, Table of n, a(n) for n = 0..10000
M. Kaneko and D. Zagier, Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998

Cf. A000005, 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).

with(numtheory); E:=proc(k) series(1-(2*k/bernoulli(k))*add( sigma[k-1](n)*q^n, n=1..60),q,61); end; E(1);

terms = 61; CoefficientList[1+4*Sum[x^k/(1-x^k), {k, 1, terms}]+O[x]^terms, x] (* Jean-François Alcover, Feb 27 2018 *)

{a(n) = if( n<1, n==0, 4 * numdiv(n))} /* Michael Somos, Jul 04 2011 */

a(0) = 1; for n >= 1, a(n) = 4*A000005(n). [After the PARI-program of Michael Somos.] - Antti Karttunen, May 25 2017

A282401 Eisenstein series E_28(q) (alternate convention E_14(q)), multiplied by 3392780147.

Original entry on oeis.org

3392780147, 6960, 934155393840, 53074158495516480, 125380214560150002480, 51856040954589843756960, 7123493021854278627673920, 457358042050198589771226240, 16828247534415852672059972400, 404722169541211889603611092720

Offset: 0

Seiichi Manyama, Feb 14 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000
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), A029831 (236364091*E_24), A282356 (657931*E_26), this sequence (3392780147*E_28).

Cf. A282402 (E_4^7), A282403 (E_4^4*E_6^2), A282404 (E_4*E_6^4).

terms = 10;
E28[x_] = 3392780147 + 6960*Sum[k^27*x^k/(1 - x^k), {k, 1, terms}];
E28[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

a(n) = 489693897*A282402(n) + 2507636250*A282403(n) + 395450000*A282404(n).

A288472 Numerators of coefficients in expansion of E_14/E_12.

Original entry on oeis.org

1, -82104, -181275671592, 1327007921039904, 16726528971891002133912, -212292443057353273999454544, -1528649681810950691089095375538848, 27164473060529924968213209402868250688, 139687438912977894660348148674573721130447640

Offset: 0

Seiichi Manyama, Jun 21 2017

			E_14/E_12 = 1 - 82104/691 * q - 181275671592/477481 * q^2  + 1327007921039904/329939371 * q^3 + 16726528971891002133912/227988105361 * q^4 + ... .

Seiichi Manyama, Table of n, a(n) for n = 0..182

Cf. A288989 (denominators).

Cf. A029828, A058550 (E_14).

terms = 9;
E14[x_] = 1 - 24*Sum[k^13*x^k/(1 - x^k), {k, 1, terms}];
E12[x_] = 1 + (65520/691)*Sum[k^11*x^k/(1 - x^k), {k, 1, terms}];
E14[x]/E12[x] + O[x]^terms // CoefficientList[#, x]& // Numerator (* Jean-François Alcover, Feb 26 2018 *)

A290009 Coefficients in expansion of 691E_4E_8*E_12.

Original entry on oeis.org

691, 563040, 305307360, 131729109120, 34085393629920, 4587384326495040, 302027782271806080, 10484303481804821760, 226150164335242994400, 3395290157453914541280, 38308806132696980919360, 343030311387007824977280, 2537869275676057371269760

Offset: 0

Seiichi Manyama, Jul 17 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A004009 (E_4), A008410 (E_8), A008411 (E_4^3), A029828 (691*E_12).

Cf. A290010.

terms = 13;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E12[x_] = 1 + (65520/691)*Sum[k^11*x^k/(1 - x^k), {k, 1, terms}];
691*E4[x]^3*E12[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

G.f.: 691*E_4^3*E_12.

cp's OEIS Frontend

A029830 Eisenstein series E_20(q) (alternate convention E_10(q)), multiplied by 174611.

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A288989 Denominators of coefficients in expansion of E_14/E_12.

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A279892 Eisenstein series E_18(q) (alternate convention E_9(q)), multiplied by 43867.

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A279893 Eisenstein series E_22(q) (alternate convention E_11(q)), multiplied by 77683.

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A282356 Eisenstein series E_26(q) (alternate convention E_13(q)), multiplied by 657931.

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A056947 Theta series of nonexistent Niemeier lattice of Coxeter number 1.

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Sage

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A145154 Coefficients in expansion of Eisenstein series E_1.

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A282401 Eisenstein series E_28(q) (alternate convention E_14(q)), multiplied by 3392780147.

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A290009 Coefficients in expansion of 691E_4E_8*E_12.