A058550 -id:A058550 - cp's OEIS frontend

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

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).

A282548 Expansion of phi_{12, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Original entry on oeis.org

0, 1, 4098, 531444, 16785412, 244140630, 2177857512, 13841287208, 68753047560, 282431130813, 1000488301740, 3138428376732, 8920506494928, 23298085122494, 56721594978384, 129747072969720, 281612482805776, 582622237229778, 1157402774071674

Offset: 0

Seiichi Manyama, Feb 18 2017

Multiplicative because A013959 is. - Andrew Howroyd, Jul 25 2018

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

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), A282050 (phi_{6, 1}), A282060 (phi_{8, 1}), A282254 (phi_{10, 1}), this sequence (phi_{12, 1}).
Cf. A282549 (E_2*E_4^3), A282576 (E_2*E_6^2), A058550 (E_14).
Cf. A013670.

Table[n * DivisorSigma[11, n], {n, 0, 18}] (* Amiram Eldar, Sep 06 2023 *)

a(n) = if(n < 1, 0, n*sigma(n, 11)) \\ Andrew Howroyd, Jul 25 2018

a(n) = n*A013959(n) for n > 0.
a(n) = (441*A282549(n) + 250*A282576(n) - 691*A058550(n))/65520.
Sum_{k=1..n} a(k) ~ zeta(12) * n^13 / 13. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(11*e+11)-1)/(p^11-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-12). (End)

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 *)

A282182 Eisenstein series E_30(q) (alternate convention E_15(q)), multiplied by 1723168255201.

Original entry on oeis.org

1723168255201, -171864, -92268782591832, -11795091175438423776, -49536425459206569762648, -32012164592742919922046864, -6332441368275869747902027488, -553385882817076320573218661312, -26594665913504249904864455466840

Offset: 0

Seiichi Manyama, Feb 16 2017

sign

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), A279893 (77683*E_22), A029831 (236364091*E_24), A282356 (657931*E_26), A282401 (3392780147*E_28), this sequence (1723168255201*E_30).

Cf. A282382 (E_4^6*E_6), A282461 (E_4^3*E_6^3), A282433 (E_6^5).

terms = 9;
E30[x_] = 1723168255201 - 171864*Sum[k^29*x^k/(1 - x^k), {k, 1, terms}];
E30[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

a(n) = 815806500201*A282382(n) + 881340705000*A282461(n) + 26021050000*A282433(n).

A282543 Coefficients in q-expansion of E_4^2*E_6^4, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Original entry on oeis.org

1, -1536, 551808, 163854336, -93387735168, -9709554816000, 4142226444876288, 642510156233453568, 41792421673548259200, 1615606968766288470528, 42343208407470359036160, 812663841518551604717568, 12060089370317565140003328

Offset: 0

Seiichi Manyama, Feb 17 2017

sign

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

Cf. A008410 (E_4^2 = E_8), A058550 (E_4^2*E_6 = E_14), A282292 (E_4^2*E_6^2 = E_10^2), A282357 (E_4^2*E_6^3), this sequence (E_4^2*E_6^4).

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]^2*E6[x]^4 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A295817 Coefficients in expansion of E_14^(-1/4).

Original entry on oeis.org

1, 6, 49248, 11042304, 6770802642, 2705631701472, 1359219630420288, 633774007586896896, 312343963839774306864, 152751427857668869125990, 75972914003765783253275712, 37915118574439727639476081152, 19063775719322131645175269693920

Offset: 0

Seiichi Manyama, Feb 13 2018

nonn

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

Cf. A058550 (E_14), A289391.

Convolution inverse of A289391.

a(n) ~ 2^(7/4) * Gamma(3/4)^9 * exp(2*Pi*n) / (3 * Pi^3 * n^(3/4)). - Vaclav Kotesovec, Jun 03 2018

A280021 Expansion of phi_{11, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Original entry on oeis.org

0, 1, 2052, 177156, 4202512, 48828150, 363524112, 1977326792, 8606744640, 31382654013, 100195363800, 285311670732, 744500215872, 1792160394206, 4057474577184, 8650199741400, 17626613022976, 34271896307922, 64397206034676, 116490258898580, 205200886312800

Offset: 0

Seiichi Manyama, Feb 22 2017

Multiplicative because A013957 is. - Andrew Howroyd, Jul 23 2018

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

Cf. A282097 (phi_{3, 2}), A282099 (phi_{5, 2}), A282751 (phi_{7, 2}), A282753 (phi_{9, 2}), this sequence (phi_{11, 2}).
Cf. A282549 (E_2*E_4^3), A282792 (E_2^2*E_4*E_6), A282576 (E_2*E_6^2), A058550 (E_4^2*E_6 = E_14).
Cf. A013957 (sigma_9(n)), A282254 (n*sigma_9(n)), this sequence (n^2*sigma_9(n)).
Cf. A013668 (zeta(10)).

Table[If[n>0, n^2 * DivisorSigma[9, n], 0], {n, 0, 20}] (* Indranil Ghosh, Mar 12 2017 *)

for(n=0, 20, print1(if(n==0, 0, n^2 * sigma(n, 9)),", ")) \\ Indranil Ghosh, Mar 12 2017

a(n) = n^2*A013957(n) for n > 0.
a(n) = (6*A282549(n) - 5*A282792(n) + 4*A282576(n) - 5*A058550(n))/1728.
Sum_{k=1..n} a(k) ~ zeta(10) * n^12 / 12. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^(2*e) * (p^(9*e+9)-1)/(p^9-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-11). (End)

A282540 Eisenstein series E_32(q) (alternate convention E_16(q)), multiplied by 7709321041217.

Original entry on oeis.org

7709321041217, 32640, 70093866303360, 20160859654708062720, 150525431711563807489920, 151991844177246093750032640, 43295116458269350559666465280, 5149788469617367127914995164160, 323250903208723929093223124860800

Offset: 0

Seiichi Manyama, Feb 17 2017

nonn

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), A279893 (77683*E_22), A029831 (236364091*E_24), A282356 (657931*E_26), A282401 (3392780147*E_28), A282182 (1723168255201*E_30), this sequence (7709321041217*E_32).

Cf. A282474 (E_4^8), A282541 (E_4^5*E_6^2), A282543 (E_4^2*E_6^4).

terms = 9;
E32[x_] = 7709321041217 + 32640*Sum[k^31*x^k/(1 - x^k), {k, 1, terms}];
E32[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

a(n) = 764412173217*A282474(n) + 5323905468000 * A282541(n) + 1621003400000 * A282543(n).

cp's OEIS Frontend

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

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A282548 Expansion of phi_{12, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

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A288472 Numerators of coefficients in expansion of E_14/E_12.

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A282182 Eisenstein series E_30(q) (alternate convention E_15(q)), multiplied by 1723168255201.

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A282543 Coefficients in q-expansion of E_4^2*E_6^4, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

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A295817 Coefficients in expansion of E_14^(-1/4).

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A280021 Expansion of phi_{11, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

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A282540 Eisenstein series E_32(q) (alternate convention E_16(q)), multiplied by 7709321041217.

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