A280022
Expansion of phi_{5, 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, 48, 324, 1792, 3750, 15552, 19208, 61440, 85293, 180000, 175692, 580608, 399854, 921984, 1215000, 2031616, 1503378, 4094064, 2606420, 6720000, 6223392, 8433216, 6716184, 19906560, 12109375, 19192992, 21257640, 34420736, 21218430, 58320000, 29552672
Offset: 0
Cf. this sequence (phi_{5, 4}),
A280025 (phi_{7, 4}).
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Table[n^4 * DivisorSigma[1, n], {n, 0, 32}] (* Amiram Eldar, Oct 31 2023 *)
nmax = 30; CoefficientList[Series[Sum[k^4*x^k*(1 + 26*x^k + 66*x^(2*k) + 26*x^(3*k) + x^(4*k))/(1 - x^k)^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 02 2025 *)
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a(n) = if(n < 1, 0, n^4 * sigma(n)); \\ Andrew Howroyd, Jul 23 2018
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.
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
a(6) = 1^6*6^3 + 2^6*3^3 + 3^6*2^3 + 6^6*1^3 = 54432.
Cf.
A282211 (phi_{4, 3}), this sequence (phi_{6, 3}).
Cf.
A001158 (sigma_3(n)),
A281372 (n*sigma_3(n)),
A282099 (n^2*sigma_3(n)), this sequence (n^3*sigma_3(n))
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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 *)
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a(n) = if (n, n^3*sigma(n, 3), 0); \\ Michel Marcus, Feb 27 2018
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
Cf.
A001160 (sigma_5(n)),
A282050 (n*sigma_5(n)), this sequence (n^2*sigma_5(n)).
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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 *)
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a(n) = if(n < 1, 0, n^2*sigma(n, 5)) \\ Andrew Howroyd, Jul 25 2018
A282780
Coefficients in q-expansion of E_2^3*E_6, where E_2 and E_6 are respectively the Eisenstein series A006352 and A013973.
Original entry on oeis.org
1, -576, 21168, 308736, -15034608, -39208320, 1590712128, 20299281408, 137107250640, 665776675008, 2599125524640, 8637331788288, 25350641846208, 67336913702016, 164742803455104, 376186503674880, 809848148403024, 1657081821679488, 3243133560510576
Offset: 0
Cf.
A282096 (E_2*E_6),
A282595 (E_2^2*E_6), this sequence (E_2^3*E_6).
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terms = 19;
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]^3*E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)
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