A013974 -id:A013974 - cp's OEIS frontend

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

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

A282213 Coefficients in q-expansion of (E_2^3E_4 - 3E_2^2E_6 + 3E_2E_4^2 - E_4E_6)/3456, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 72, 756, 4672, 15750, 54432, 117992, 299520, 551853, 1134000, 1772892, 3532032, 4829006, 8495424, 11907000, 19173376, 24142482, 39733416, 47052740, 73584000, 89201952, 127648224, 148048056, 226437120, 246109375, 347688432, 402320520, 551258624, 594847710

Offset: 0

Seiichi Manyama, Feb 09 2017

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

			a(6) = 1^6*6^3 + 2^6*3^3 + 3^6*2^3 + 6^6*1^3 = 54432.

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

Cf. A282211 (phi_{4, 3}), this sequence (phi_{6, 3}).
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282586 (E_2^3*E_4), A282595 (E_2^2*E_6), A282101 (E_2*E_4^2), A013974 (E_4*E_6 = E_10).
Cf. A001158 (sigma_3(n)), A281372 (n*sigma_3(n)), A282099 (n^2*sigma_3(n)), this sequence (n^3*sigma_3(n))
Cf. A013662.

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}];
(E2[x]^3*E4[x] - 3 E2[x]^2*E6[x] + 3 E2[x] E4[x]^2 - E4[x] E6[x])/3456 + O[x]^terms // CoefficientList[#, x]&
(* or: *)
Table[n^3*DivisorSigma[3, n], {n, 0, terms-1}] (* Jean-François Alcover, Feb 27 2018 *)
nmax = 30; CoefficientList[Series[Sum[k^6*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 02 2025 *)

a(n) = if (n, n^3*sigma(n, 3), 0); \\ Michel Marcus, Feb 27 2018

G.f.: phi_{6, 3}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
a(n) = (A282586(n) - 3*A282595(n) + 3*A282101(n) - A013974(n))/3456. - Seiichi Manyama, Feb 19 2017
a(n) = n^3*A001158(n) for n > 0. - Seiichi Manyama, Feb 19 2017
Sum_{k=1..n} a(k) ~ zeta(4) * n^7 / 7. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 31 2023: (Start)
Multiplicative with a(p^e) = p^(3*e) * (p^(3*e+3)-1)/(p^3-1).
Dirichlet g.f.: zeta(s-3)*zeta(s-6). (End)
G.f.: Sum_{k>=1} k^6*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4. - Vaclav Kotesovec, Aug 02 2025

A282751 Expansion of phi_{7, 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, 132, 2196, 16912, 78150, 289872, 823592, 2164800, 4802733, 10315800, 19487292, 37138752, 62748686, 108714144, 171617400, 277094656, 410338962, 633960756, 893872100, 1321672800, 1808608032, 2572322544, 3404825976, 4753900800, 6105469375, 8282826552

Offset: 0

Seiichi Manyama, Feb 21 2017

Multiplicative because A001160 is. - Andrew Howroyd, Jul 25 2018

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

Cf. A282097 (phi_{3, 2}), A282099 (phi_{5, 2}), this sequence (phi_{7, 2}), A282753 (phi_{9, 2}).
Cf. A282101 (E_2*E_4^2), A282595 (E_2^2*E_6), A013974 (E_4*E_6 = E_10).
Cf. A001160 (sigma_5(n)), A282050 (n*sigma_5(n)), this sequence (n^2*sigma_5(n)).
Cf. A013664.

Table[n^2 * DivisorSigma[5, n], {n, 0, 30}] (* Amiram Eldar, Sep 06 2023 *)
nmax = 40; CoefficientList[Series[Sum[k^7*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 < 1, 0, n^2*sigma(n, 5)) \\ Andrew Howroyd, Jul 25 2018

a(n) = n^2*A001160(n) for n > 0.
a(n) = (2*A282101(n) - A282595(n) - A013974(n))/1728.
Sum_{k=1..n} a(k) ~ zeta(6) * n^8 / 8. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^(2*e) * (p^(5*e+5)-1)/(p^5-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-7). (End)
G.f.: Sum_{k>=1} k^7*x^k*(1 + x^k)/(1 - x^k)^3. - Vaclav Kotesovec, Aug 02 2025

A289294 Coefficients in expansion of E_10^(1/2).

Original entry on oeis.org

1, -132, -76428, -12686784, -4629945804, -1581036186312, -643032851554368, -264454897726360704, -114830224962140965068, -50847479367845783084484, -23070238839261012248537688, -10629338992044523324726971456

Offset: 0

Seiichi Manyama, Jul 02 2017

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

E_k^(1/2): A289291 (k=2), A289292 (k=4), A289293 (k=6), A004009 (k=8), this sequence (k=10), A289295 (k=14).

Cf. A013974 (E_10), A289024.

nmax = 20; s = 10; CoefficientList[Series[Sqrt[1 - 2*s/BernoulliB[s] * Sum[DivisorSigma[s - 1, k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 02 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(A289024(n)/2).

a(n) ~ c * exp(2*Pi*n) / n^(3/2), where c = -3^(3/2) * Pi^(5/2) / (2^(9/2) * Gamma(3/4)^12) = -0.3503612261281732359954402284478780636268623476628... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018

A282000 Coefficients in q-expansion of E_4^3*E_6, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Original entry on oeis.org

1, 216, -200232, -85500576, -11218984488, -499862636784, -11084671590048, -152346382155072, -1474691273530920, -10921720940625672, -65489246355989232, -331011680696545248, -1452954445366288032, -5665058572086302256, -19968589327695656256

Offset: 0

Seiichi Manyama, Feb 05 2017

sign

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), A013973 (E_6), A013974 (E_4*E_6 = E_10), A058550 (E_4^2*E_6 = E_14), this sequence (E_4^3*E_6), A282047 (E_4^4*E_6), A282048 (E_4^5*E_6).

Cf. A013965, A037944, A281928.

terms = 15;
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] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

-28728 * A013965(n) = 43867 * a(n) - 9504000 * A037944(n) for n > 0.

A285836 Coefficients in expansion of 1/E_10.

Original entry on oeis.org

1, 264, 205128, 95104416, 54329698632, 28308006715824, 15339873507244704, 8172566140980183360, 4385988806258507934024, 2346434028637391065282536, 1257009611855633134427201328, 672999598306502464042506285792

Offset: 0

Seiichi Manyama, Jun 17 2017

nonn

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

Cf. A013974 (E_10).

Cf. A288816 (k=2), A001943 (k=4), A000706 (k=6), A287933 (k=8), this sequence (k=10), A287964 (k=14).

a(n) ~ c * exp(2*Pi*n), where c = 128 * Gamma(3/4)^24 / (27 * Pi^6) = 0.648273189440897942951926047466605067667211940159693598407336163991191821438... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018

A290181 Coefficients in expansion of E_10*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

Original entry on oeis.org

1, -312, -121680, 1004000, 37942020, -801594864, 6139193600, -11831002560, -151614128250, 1346611783000, -4592794000704, 3738595861728, 15192491492360, 47281379454000, -737660590018560, 2662090686805056, -3290770281735027, -4884703150768920

Offset: 2

Seiichi Manyama, Jul 23 2017

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Seiichi Manyama, Table of n, a(n) for n = 2..1000
S. C. Milne, Hankel determinants of Eisenstein series, preprint, arXiv:0009130 [math.NT], 2000.

E_k*Delta^2: A290178 (k=4), A290048 (k=6), A290180 (k=8), this sequence (k=10), A290182 (k=14).

Cf. A000594, A013974 (E_10).

terms = 18;
E10[x_] = 1 - 264*Sum[k^9*x^k/(1 - x^k), {k, 1, terms}];
E10[x]*QPochhammer[x]^48 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Let b(q) be the determinant of the 3 X 3 matrix [E_6, E_10, E_12 ; E_8, E_12, E_14 ; E_10, E_14, E_16]. G.f. is 691^2*3617*b(q)/(1728^2*2^2*3*5^6*7^2*13).

A282047 Coefficients in q-expansion of E_4^4*E_6, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Original entry on oeis.org

1, 456, -146232, -133082976, -32170154808, -3378441902544, -155862776255328, -3969266446940352, -65538944782146360, -777506848190979672, -7105808014591457232, -52584752452485047328, -326903300701760852832, -1755591608260377411216

Offset: 0

Seiichi Manyama, Feb 05 2017

sign

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), A013973 (E_6), A013974 (E_4*E_6 = E_10), A058550 (E_4^2*E_6 = E_14), A282000 (E_4^3*E_6), this sequence (E_4^4*E_6), A282048 (E_4^5*E_6).

Cf. A013969, A037946, A281956.

terms = 14;
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]^4*E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

-552 * A013969(n) = 77683 * a(n) - 35424000 * A037946(n) for n > 0.

A282048 Coefficients in q-expansion of E_4^5*E_6, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Original entry on oeis.org

1, 696, -34632, -167186976, -64422848328, -11387712944304, -1037073232984608, -48892286706157632, -1378097272692189000, -26188038166214133672, -364779879415169299632, -3952277018332870144608, -34798618196377082329632, -257403706082325167732976

Offset: 0

Seiichi Manyama, Feb 05 2017

sign

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), A013973 (E_6), A013974 (E_4*E_6 = E_10), A058550 (E_4^2*E_6 = E_14), A282000 (E_4^3*E_6), A282047 (E_4^4*E_6), this sequence (E_4^5*E_6).

Cf. A037947, A281959, A281979.

terms = 14;
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]^5*E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

-24 * A281959(n) = 657931 * a(n) - 457920000 * A037947(n) for n > 0.

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).

cp's OEIS Frontend

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

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A282213 Coefficients in q-expansion of (E_2^3*E_4 - 3*E_2^2*E_6 + 3*E_2*E_4^2 - E_4*E_6)/3456, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A282751 Expansion of phi_{7, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

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A289294 Coefficients in expansion of E_10^(1/2).

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A282000 Coefficients in q-expansion of E_4^3*E_6, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

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A285836 Coefficients in expansion of 1/E_10.

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A290181 Coefficients in expansion of E_10*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

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A282047 Coefficients in q-expansion of E_4^4*E_6, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

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A282048 Coefficients in q-expansion of E_4^5*E_6, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

A282213 Coefficients in q-expansion of (E_2^3E_4 - 3E_2^2E_6 + 3E_2E_4^2 - E_4E_6)/3456, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.