A282102 -id:A282102 - cp's OEIS frontend

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.

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

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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 *)

A282287 Coefficients in q-expansion of E_4*E_6^2, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Original entry on oeis.org

1, -768, -19008, 67329024, 4834170816, 137655866880, 2122110676224, 21418943158272, 158760815970240, 928988742914304, 4512155542392960, 18847838706545664, 69519052583699712, 230952254655327744, 701948326302761472, 1975789128222443520

Offset: 0

Seiichi Manyama, Feb 11 2017

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

Cf. A004009 (E_4), A013973 (E_6), A008411 (E_4^3), A058550 (E_4^2*E_6 = E_14), this sequence (E_4*E_6^2), A282253 (E_6^3).

Cf. A282102 (E_2*E_10), A058550 (E_4*E_10), this sequence (E_6*E_10).

terms = 16;
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]*E6[x]^2 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

A280025 Expansion of phi_{7, 4}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Original entry on oeis.org

0, 1, 144, 2268, 18688, 78750, 326592, 825944, 2396160, 4966677, 11340000, 19501812, 42384384, 62777078, 118935936, 178605000, 306774016, 410422194, 715201488, 894002060, 1471680000, 1873240992, 2808260928, 3405105288, 5434490880, 6152734375, 9039899232

Offset: 0

Seiichi Manyama, Feb 22 2017

Multiplicative because A001158 is. - Andrew Howroyd, Jul 23 2018

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

Cf. A280022 (phi_{5, 4}), this sequence (phi_{7, 4}).
Cf. A280024 (E_2^4*E_4), A282780 (E_2^3*E_6), A282752 (E_2^2*E_4^2), A282102 (E_2*E_4*E_6), A008411 (E_4^3), A280869 (E_6^2).
Cf. A001158 (sigma_3(n)), A281372 (n*sigma_3(n)), A282099 (n^2*sigma_3(n)), A282213 (n^3*sigma_3(n)), this sequence (n^4*sigma_3(n)).
Cf. A152649.

Table[n^4 * DivisorSigma[3, n], {n, 0, 30}] (* Amiram Eldar, Oct 31 2023 *)
nmax = 30; CoefficientList[Series[Sum[k^4*x^k*(1 + 120*x^k + 1191*x^(2*k) + 2416*x^(3*k) + 1191*x^(4*k) + 120*x^(5*k) + x^(6*k))/(1 - x^k)^8, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 02 2025 *)

a(n) = if(n < 1, 0, n^4 * sigma(n, 3)); \\ Andrew Howroyd, Jul 23 2018

a(n) = n^4*A001158(n) for n > 0.
a(n) = (7*(A280024(n) - 4*A282780(n) + 6*A282752(n) - 4*A282102(n)) + 3*A008411(n) + 4*A280869(n))/41472.
Sum_{k=1..n} a(k) ~ c * n^8, where c = Pi^4/720 = 0.1352904... (= A152649 / 10). - Amiram Eldar, Dec 08 2022
From Amiram Eldar, Oct 31 2023: (Start)
Multiplicative with a(p^e) = p^(4*e) * (p^(3*e+3)-1)/(p^3-1).
Dirichlet g.f.: zeta(s-4)*zeta(s-7). (End)
G.f.: Sum_{k>=1} k^4*x^k*(1 + 120*x^k + 1191*x^(2*k) + 2416*x^(3*k) + 1191*x^(4*k) + 120*x^(5*k) + x^(6*k))/(1 - x^k)^8. - Vaclav Kotesovec, Aug 02 2025

A282096 Coefficients in q-expansion of E_2*E_6, where E_2, E_6 are the Eisenstein series shown in A006352, A013973, respectively.

Original entry on oeis.org

1, -528, -4608, 312384, 3664416, 21745440, 86782464, 276703872, 741794400, 1758969264, 3797729280, 7568097984, 14222957952, 25253852064, 43166426112, 70518360960, 112406614752, 172631876832, 260795119104, 381636168000, 552633117120, 778105665024

Offset: 0

Seiichi Manyama, Feb 06 2017

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

Cf. A006352 (E_2), A013973 (E_6).

Cf. A281374 (E_2^2), A282019 (E_2*E_4), this sequence (E_2*E_6), A282101 (E_2*E_8), A282102 (E_2*E_10).

terms = 22;
E2[x_] = 1 - 24*Sum[k*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]*E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

A282254 Coefficients in q-expansion of (3E_4^3 + 2E_6^2 - 5E_2E_4*E_6)/1584, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 1026, 59052, 1050628, 9765630, 60587352, 282475256, 1075843080, 3486961557, 10019536380, 25937424612, 62041684656, 137858491862, 289819612656, 576679982760, 1101663313936, 2015993900466, 3577622557482, 6131066257820, 10260044315640

Offset: 0

Seiichi Manyama, Feb 10 2017

Multiplicative because A013957 is. - Andrew Howroyd, Jul 25 2018

D. H. Lehmer shows that a(n) == tau(n) (mod 7) for n > 0, where tau is Ramanujan's tau function (A000594). Furthermore, if n == 3, 5, 6 (mod 7) then a(n) == tau(n) (mod 49). See the Wikipedia link below. - Jianing Song, Aug 12 2020

			a(6) = 1^10*6^1 + 2^10*3^1 + 3^10*2^1 + 6^10*1^1 = 60587352.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Wikipedia, Congruences for the tau function.

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), A282050 (phi_{6, 1}), A282060 (phi_{8, 1}), this sequence (phi_{10, 1}).
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A008411 (E_4^3), A280869 (E_6^2), A282102 (E_2*E_4*E_6).
Cf. A013957, A013668, A196292.

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

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

G.f.: phi_{10, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
a(n) = (3*A008411(n) + 2*A280869(n) - 5*A282102(n))/1584.
If p is a prime, a(p) = p^10 + p = A196292(p).
a(n) = n*A013957(n) for n > 0, where A013957(n) is sigma_9(n), the sum of the 9th powers of the divisors of n. - Seiichi Manyama, Feb 18 2017
Multiplicative with a(p^e) = p^e*(p^(9*(e+1))-1)/(p^9-1). - Jianing Song, Aug 12 2020
From Amiram Eldar, Oct 30 2023: (Start)
Dirichlet g.f.: zeta(s-1)*zeta(s-10).
Sum_{k=1..n} a(k) ~ zeta(10) * n^11 / 11. (End)

A282753 Expansion of phi_{9, 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, 516, 19692, 264208, 1953150, 10161072, 40353656, 135274560, 387597717, 1007825400, 2357947812, 5202783936, 10604499542, 20822486496, 38461429800, 69260574976, 118587876786, 200000421972, 322687698140, 516037855200, 794644193952, 1216701070992

Offset: 0

Seiichi Manyama, Feb 21 2017

Multiplicative because A013955 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}), A282751 (phi_{7, 2}), this sequence (phi_{9, 2}).
Cf. A282752 (E_2^2*E_4^2), A282102 (E_2*E_4*E_6), A008411 (E_4^3), A280869 (E_6^2).
Cf. A013955 (sigma_7(n)), A282060 (n*sigma_7(n)), this sequence (n^2*sigma_7(n)).
Cf. A013666.

Table[If[n>0, n^2 * DivisorSigma[7, n], 0], {n, 0, 22}] (* Indranil Ghosh, Mar 12 2017 *)
nmax = 40; CoefficientList[Series[Sum[k^9*x^k*(1 + x^k)/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 02 2025 *)

for(n=0, 22, print1(if(n==0, 0, n^2 * sigma(n, 7)),", ")) \\ Indranil Ghosh, Mar 12 2017

a(n) = n^2*A013955(n) for n > 0.
a(n) = (9*A282752(n) - 18*A282102(n) + 5*A008411(n) + 4*A280869(n))/8640.
Sum_{k=1..n} a(k) ~ zeta(8) * n^10 / 10. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^(2*e) * (p^(7*e+7)-1)/(p^7-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-9). (End)
G.f.: Sum_{k>=1} k^9*x^k*(1 + x^k)/(1 - x^k)^3. - Vaclav Kotesovec, Aug 02 2025

A282596 Coefficients in q-expansion of E_2E_4^2E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

1, -48, -196128, -33542976, -678319104, 12136422240, 509314518144, 7469015889792, 68272650653760, 458377820557584, 2454769903187520, 11035857376010304, 43103740076823552, 149954656815201504, 473331019057949952, 1375248429330791040, 3719662610125117632

Offset: 0

Seiichi Manyama, Feb 19 2017

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

Cf. A282102 (E_2*E_4*E_6), A282547 (E_2*E_4*E_6^2).

terms = 17;
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]* E4[x]^2 *E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A282781 Expansion of phi_{8, 3}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Original entry on oeis.org

0, 1, 264, 6588, 67648, 390750, 1739232, 5765144, 17318400, 43224597, 103158000, 214360212, 445665024, 815732918, 1521998016, 2574261000, 4433514496, 6975762354, 11411293608, 16983569900, 26433456000, 37980768672, 56591095968, 78310997448

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. A282211 (phi_{4, 3}), A282213 (phi_{6, 3}), this sequence (phi_{8, 3}).
Cf. A282752 (E_2^2*E_4^2), A282780 (E_2^3*E_6), A282102 (E_2*E_4*E_6), A008411 (E_4^3), A280869 (E_6^2).
Cf. A001160 (sigma_5(n)), A282050 (n*sigma_5(n)), A282751 (n^2*sigma_5(n)), this sequence (n^3*sigma_5(n)).
Cf. A013664.

a[0]=0;a[n_]:=(n^3)*DivisorSigma[5,n];Table[a[n],{n,0,23}] (* Indranil Ghosh, Feb 21 2017 *)
nmax = 30; CoefficientList[Series[Sum[k^8*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==0, 0, n^3*sigma(n, 5)); \\ Michel Marcus, Feb 21 2017

a(n) = n^3*A001160(n) for n > 0.
a(n) = (6*A282752(n) - 2*A282780(n) - 6*A282102(n) + A008411(n) + A280869(n))/5184.
Sum_{k=1..n} a(k) ~ zeta(6) * n^9 / 9. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 31 2023: (Start)
Multiplicative with a(p^e) = p^(3*e) * (p^(5*e+5)-1)/(p^5-1).
Dirichlet g.f.: zeta(s-3)*zeta(s-8). (End)
G.f.: Sum_{k>=1} k^8*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4. - Vaclav Kotesovec, Aug 02 2025

A282792 Coefficients in q-expansion of E_2^2E_4E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

1, -312, -122328, 1193376, 120735336, 123318576, -26119268064, -383848045248, -3132125965080, -18290795499096, -84925855577232, -331983655889184, -1133781877844448, -3470165144567184, -9697162366507968, -25093220330304576, -60786860467926552

Offset: 0

Seiichi Manyama, Feb 21 2017

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

Cf. A282102 (E_2*E_4*E_6), this sequence (E_2^2*E_4*E_6), A282596 (E_2*E_4^2*E_6), A282547 (E_2*E_4*E_6^2).

terms = 17;
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*E6[x]*E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A126861 Coefficients in quasimodular form 12*F_3(q) of level 1 and weight 12.

Original entry on oeis.org

0, 0, 1, 80, 1224, 9152, 45276, 170784, 534464, 1438848, 3507102, 7711600, 16053728, 30831552, 57578072, 100382304, 173117952, 280579200, 455656725, 697508496, 1079398256, 1580599552, 2351610612, 3315523424, 4785293568, 6534524160, 9173253878, 12226860576

Offset: 0

N. J. A. Sloane, Mar 15 2007

nonn

			12*F_3(q) = q^2 + 80*q^3 + 1224*q^4 + 9152*q^5 + 45276*q^6 + 170784*q^7 + 534464*q^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
B. Mazur, Perturbations, deformations and variations (and "near-misses") in geometry, physics, and number theory, Bull. Amer. Math. Soc., 41 (2004), 307-336.

Cf. A126858, A126859, A126860.

Cf. A280024 (E_2^4*E_4), A308285 (E_2^6), A282752 (E_2^2*E_4^2), A008411 (E_4^3), A282780 (E_2^3*E_6), A282102 (E_2*E_4*E_6), A280869 (E_6^2).

F_3(q) = (15*E(2)^4*E(4) - 6*E(2)^6 - 12*E(2)^2*E(4)^2 + 7*E(4)^3 + 4*E(2)^3*E(6) - 12*E(2)*E(4)*E(6) + 4*E(6)^2)/35831808, where E(k) is the normalized Eisenstein series of weight k (cf. A006352, etc.).

cp's OEIS Frontend

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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A282287 Coefficients in q-expansion of E_4*E_6^2, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

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A280025 Expansion of phi_{7, 4}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

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A282096 Coefficients in q-expansion of E_2*E_6, where E_2, E_6 are the Eisenstein series shown in A006352, A013973, respectively.

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A282254 Coefficients in q-expansion of (3*E_4^3 + 2*E_6^2 - 5*E_2*E_4*E_6)/1584, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A282753 Expansion of phi_{9, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

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A282596 Coefficients in q-expansion of E_2*E_4^2*E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A282781 Expansion of phi_{8, 3}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

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A282792 Coefficients in q-expansion of E_2^2*E_4*E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A282254 Coefficients in q-expansion of (3E_4^3 + 2E_6^2 - 5E_2E_4*E_6)/1584, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

A282596 Coefficients in q-expansion of E_2E_4^2E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

A282792 Coefficients in q-expansion of E_2^2E_4E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.