A008410 -id:A008410 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

A288840 Coefficients in expansion of E_8/E_6.

Original entry on oeis.org

1, 984, 574488, 307081056, 164453203992, 88062998451984, 47157008244215904, 25252184242734325440, 13522333949728177520664, 7241096993206804017918456, 3877547016709833498690361488, 2076394071353012138642420600352

Offset: 0

Seiichi Manyama, Jun 17 2017

nonn

			G.f.: 1 + 984*q + 574488*q^2 + 307081056*q^3 + 164453203992*q^4 + 88062998451984*q^5 + 47157008244215904*q^6 + ...
From _Seiichi Manyama_, Jun 27 2017: (Start)
a(0) = j_0(i) = 1,_
a(1) = j_1(i) = -744 + 1728^1 = 984,
a(2) = j_2(i) = 159768 - 1488*1728^1 + 1728^2 = 574488. (End)

Ken Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-series, CBMS Regional Conference Series in Mathematics, vol. 102, American Mathematical Society, Providence, RI, 2004.

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

Cf. A013973 (E_6), A008410 (E_8).
Cf. A288261 (E_6/E_4).
Cf. A000521 (j), A035230 (-q*j'), A289141, A289417.

nmax = 20; CoefficientList[Series[(1 + 480*Sum[DivisorSigma[7, k]*x^k, {k, 1, nmax}])/(1 - 504*Sum[DivisorSigma[5, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 28 2017 *)
terms = 12; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[Ei[8]/Ei[6] + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)

From Seiichi Manyama, Jun 27 2017: (Start)
Let j_0 = 1 and j_1 = j - 744. Define j_m by j_m = j1 | T_0(m), where T_0(m) = mT_{m, 0} is the normalized m-th weight zero Hecke operator. a(n) = j_n(i).
G.f.: Sum_{n >= 0} j_n(i)*q^n. (End)
a(n) ~ 2 * exp(2*Pi*n). - Vaclav Kotesovec, Jun 28 2017
G.f.: -q*j'/(j-1728) where j is the elliptic modular invariant (A000521). - Seiichi Manyama, Jul 12 2017

A282012 Coefficients in q-expansion of E_4^4, where E_4 is the Eisenstein series shown in A004009.

Original entry on oeis.org

1, 960, 354240, 61543680, 4858169280, 137745912960, 2120861041920, 21423820362240, 158753769048000, 928983317334720, 4512174992346240, 18847874280625920, 69518972236842240, 230951926208599680, 701949379778818560, 1975788826748167680

Offset: 0

Seiichi Manyama, Feb 04 2017

nonn

Also coefficients in q-expansion of E_8^2.

G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 207.

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

Cf. A004009 (E_4), A008410 (E_4^2), A008411 (E_4^3), this sequence (E_4^4), A282015 (E_4^5).
Cf. A281374 (E_2^2), A008410 (E_4^2), A280869 (E_6^2), this sequence (E_8^2), A282292 (E_10^2).
Cf. A013963, A027364, A281876.

terms = 16;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E4[x]^4 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

G.f.: (1 + 240 Sum_{i>=1} i^3 q^i/(1-q^i))^4.

16320 * A013963(n) = 3617 * a(n) - 3456000 * A027364(n) for n > 0.

A282050 Coefficients in q-expansion of (E_4^2 - E_2*E_6)/1008, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 66, 732, 4228, 15630, 48312, 117656, 270600, 533637, 1031580, 1771572, 3094896, 4826822, 7765296, 11441160, 17318416, 24137586, 35220042, 47045900, 66083640, 86124192, 116923752, 148035912, 198079200, 244218775, 318570252, 389021400, 497449568, 594823350

Offset: 0

Seiichi Manyama, Feb 05 2017

Multiplicative because A001160 is. - Andrew Howroyd, Jul 23 2018

			a(6) = 1^6*6^1 + 2^6*3^1 + 3^6*2^1 + 6^6*1^1 = 48312.

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

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), this sequence (phi_{6, 1}), A282060 (phi_{8, 1}).
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A145095 (-q*E'_6), A008410 (E_4^2 = E_8), A282096 (E_2*E_6).
Cf. A001160, A013664, A131472.

terms = 30;
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}];
(E4[x]^2 - E2[x]*E6[x])/1008 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)
Table[n * DivisorSigma[5, n], {n, 0, 30}] (* Amiram Eldar, Sep 06 2023 *)
nmax = 40; CoefficientList[Series[x*Sum[k^6*x^(k-1)/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 01 2025 *)

a(n) = if(n < 1, 0, n * sigma(n, 5)); \\ Andrew Howroyd, Jul 23 2018

a(n) = A145095(n)/504 for n > 0.
G.f.: phi_{6, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
a(n) = (A008410(n) - A282096(n))/1008. - Seiichi Manyama, Feb 10 2017
If p is a prime, a(p) = p^6 + p = A131472(p). - Seiichi Manyama, Feb 10 2017
a(n) = n*A001160(n) for n > 0. - Seiichi Manyama, Feb 18 2017
Sum_{k=1..n} a(k) ~ zeta(6) * n^7 / 7. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(5*e+5)-1)/(p^5-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-6). (End)
G.f. Sum_{k>=1} k^6*x^(k-1)/(1 - x^k)^2. - Vaclav Kotesovec, Aug 01 2025

A282101 Coefficients in q-expansion of E_2*E_4^2, where E_2, E_4 are the Eisenstein series shown in A006352, A004009, respectively.

Original entry on oeis.org

1, 456, 50328, -470496, -21784008, -234371664, -1446514848, -6502690752, -23328111240, -71276388312, -191952331632, -468159788448, -1052750026272, -2212261706256, -4394299104576, -8303419066176, -15060718806024, -26284654025712, -44471780630856

Offset: 0

Seiichi Manyama, Feb 06 2017

sign

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

Cf. A006352 (E_2), A004009 (E_4), A008410 (E_8).

Cf. A281374 (E_2^2), A282019 (E_2*E_4), A282096 (E_2*E_6), this sequence (E_2*E_8), A282102 (E_2*E_10).

terms = 19;
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}];
E2[x]*E4[x]^2 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

A282015 Coefficients in q-expansion of E_4^5, where E_4 is the Eisenstein series shown in A004009.

Original entry on oeis.org

1, 1200, 586800, 148641600, 20400279600, 1439038231200, 46093334702400, 861697555612800, 10894180752126000, 102121497049868400, 755966260027216800, 4623420005167550400, 24151632380348692800, 110516281318431693600, 451789183426135939200

Offset: 0

Seiichi Manyama, Feb 05 2017

nonn

G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 208.

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

Cf. A004009 (E_4), A008410 (E_4^2), A008411 (E_4^3), A282012 (E_4^4), this sequence (E_4^5).

Cf. A013967, A037945, A281788.

terms = 15;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E4[x]^5 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

G.f.: (1 + 240 Sum_{i>=1} i^3 q^i/(1-q^i))^5.

13200 * A013967(n) = 174611 * a(n) - 209520000 * A037945(n) for n > 0.

A282099 Coefficients in q-expansion of (E_2^2E_4 - 2E_2*E_6 + E_4^2)/1728, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 36, 252, 1168, 3150, 9072, 16856, 37440, 61317, 113400, 161172, 294336, 371462, 606816, 793800, 1198336, 1420146, 2207412, 2476460, 3679200, 4247712, 5802192, 6436872, 9434880, 9844375, 13372632, 14900760, 19687808, 20511990, 28576800, 28630112, 38347776

Offset: 0

Seiichi Manyama, Feb 06 2017

Multiplicative because A001158 is. - Andrew Howroyd, Jul 25 2018

			a(6) = 1^5*6^2 + 2^5*3^2 + 3^5*2^2 + 6^5*1^2 = 9072.

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

Cf. A282097 (phi_{3, 2}), this sequence (phi_{5, 2}).
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282208 (E_2^2*E_4), A282096 (E_2*E_6), A008410 (E_8 = E_4^2).
Cf. A001158 (sigma_3(n)), A281372 (n*sigma_3(n)), this sequence (n^2*sigma_3(n)), A282213 (n^3*sigma_3(n)).

a[0]=0;a[n_]:=(n^2)*DivisorSigma[3,n];Table[a[n],{n,0,32}] (* Indranil Ghosh, Feb 21 2017 *)
nmax = 40; CoefficientList[Series[Sum[k^5*x^k*(1 + x^k)/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 02 2025 *)

a(n) = if (n==0, 0, n^2*sigma(n, 3)); \\ Michel Marcus, Feb 21 2017

G.f.: phi_{5, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
a(n) = (A282208(n) - 2*A282096(n) + A008410(n))/1728. - Seiichi Manyama, Feb 19 2017
a(n) = n^2*A001158(n) for n > 0. - Seiichi Manyama, Feb 19 2017
Sum_{k=1..n} a(k) ~ Pi^4 * n^6 / 540. - Vaclav Kotesovec, May 09 2022
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^(2*e) * (p^(3*e+3)-1)/(p^3-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-5). (End)
G.f.: Sum_{k>=1} k^5*x^k*(1 + x^k)/(1 - x^k)^3. - Vaclav Kotesovec, Aug 02 2025

A282211 Coefficients in q-expansion of (6E_2^2E_4 - 8E_2E_6 + 3*E_4^2 - E_2^4)/6912, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 24, 108, 448, 750, 2592, 2744, 7680, 9477, 18000, 15972, 48384, 30758, 65856, 81000, 126976, 88434, 227448, 137180, 336000, 296352, 383328, 292008, 829440, 484375, 738192, 787320, 1229312, 731670, 1944000, 953312, 2064384, 1724976

Offset: 0

Seiichi Manyama, Feb 09 2017

Multiplicative because A000203 is. - Andrew Howroyd, Jul 25 2018

			a(6) = 1^4*6^3 + 2^4*3^3 + 3^4*2^3 + 6^4*1^3 = 2592.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

Cf. this sequence (phi_{4, 3}), A282213 (phi_{6, 3}).
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282208 (E_2^2*E_4), A282096 (E_2*E_6), A008410 (E_4^2 = E_8), A282210 (E_2^4).
Cf. A000203 (sigma(n)), A064987 (n*sigma(n)), A282097 (n^2*sigma(n)), this sequence (n^3*sigma(n)).

a[0]=0;a[n_]:=(n^3)*DivisorSigma[1,n];Table[a[n],{n,0,33}] (* Indranil Ghosh, Feb 21 2017 *)

a(n) = if (n==0, 0, n^3*sigma(n)); \\ Michel Marcus, Feb 21 2017

G.f.: phi_{4, 3}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
a(n) = (6*A282208(n) - 8*A282096(n) + 3*A008410(n) - A282210(n))/6912.
a(n) = n^3*A000203(n) for n > 0. - Seiichi Manyama, Feb 19 2017
G.f.: A(q) = Sum_{n >= 1} n^3*q^n*(q^(3*n) + 11*q^(2*n) + 11*q^n + 1)/(1 - q^n)^5. A faster converging series may be found by applying the operator x*d/dx once to equation 5 in Arndt, setting x = 1, and then applying the operator q*d/dq three times to the resulting equation. - Peter Bala, Jan 21 2021
Sum_{k=1..n} a(k) ~ c * n^5, where c = Pi^2/30 = 0.328986... . - Amiram Eldar, Dec 08 2022
From Amiram Eldar, Oct 31 2023: (Start)
Multiplicative with a(p^e) = p^(3*e) * (p^(e+1)-1)/(p-1).
Dirichlet g.f.: zeta(s-3)*zeta(s-4). (End)
G.f.: A(q) = Sum_{n >= 1} n^4*q^n*(q^(2*n) + 4*q^n + 1)/(1 - q^n)^4. - Mamuka Jibladze, Aug 27 2024

A282292 Coefficients in q-expansion of E_10^2, where E_10 is the Eisenstein series A013974.

Original entry on oeis.org

1, -528, -201168, 61114944, 20946935856, 1443146395680, 46053422547264, 861726789128832, 10894843149545520, 102119072037503664, 755968133350219680, 4623420033182073024, 24151660069581371712, 110516194189880866464, 451789196756619249792

Offset: 0

Seiichi Manyama, Feb 11 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Eisenstein Series.

Cf. A281374 (E_2^2), A008410 (E_4^2), A280869 (E_6^2), A282012 (E_8^2), this sequence (E_10^2).

terms = 15;
E10[x_] = 1 - 264*Sum[k^9*x^k/(1 - x^k), {k, 1, terms}];
E10[x]^2 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

cp's OEIS Frontend

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

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

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A288840 Coefficients in expansion of E_8/E_6.

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A282012 Coefficients in q-expansion of E_4^4, where E_4 is the Eisenstein series shown in A004009.

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A282050 Coefficients in q-expansion of (E_4^2 - E_2*E_6)/1008, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A282101 Coefficients in q-expansion of E_2*E_4^2, where E_2, E_4 are the Eisenstein series shown in A006352, A004009, respectively.

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A282015 Coefficients in q-expansion of E_4^5, where E_4 is the Eisenstein series shown in A004009.

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A282099 Coefficients in q-expansion of (E_2^2*E_4 - 2*E_2*E_6 + E_4^2)/1728, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A282099 Coefficients in q-expansion of (E_2^2E_4 - 2E_2*E_6 + E_4^2)/1728, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

A282211 Coefficients in q-expansion of (6E_2^2E_4 - 8E_2E_6 + 3*E_4^2 - E_2^4)/6912, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.