A282431 -id:A282431 - cp's OEIS frontend

A280022 Expansion of phi_{5, 4}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

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

Seiichi Manyama, Feb 22 2017

Multiplicative because A000203 is. - Andrew Howroyd, Jul 23 2018

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

Cf. this sequence (phi_{5, 4}), A280025 (phi_{7, 4}).
Cf. A282101 (E_2*E_4^2), A282595 (E_2^2*E_6), A282586 (E_2^3*E_4), A013974 (E_4*E_6 = E_10), A282431 (E_2^5).
Cf. A000203 (sigma(n)), A064987 (n*sigma(n)), A282097 (n^2*sigma(n)), A282211 (n^3*sigma(n)), this sequence (n^4*sigma(n)).
Cf. A353908.

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 *)

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

a(n) = n^4*A000203(n) for n > 0.
a(n) = (15*A282101(n) - 20*A282595(n) + 10*A282586(n) - 4*A013974(n) - A282431(n))/20736.
Sum_{k=1..n} a(k) ~ c * n^6, where c = Pi^2/36 = 0.274155... (A353908). - Amiram Eldar, Dec 08 2022
From Amiram Eldar, Oct 31 2023: (Start)
Multiplicative with a(p^e) = p^(4*e) * (p^(e+1)-1)/(p-1).
Dirichlet g.f.: zeta(s-4)*zeta(s-5). (End)
G.f.: Sum_{k>=1} k^4*x^k*(1 + 26*x^k + 66*x^(2*k) + 26*x^(3*k) + x^(4*k))/(1 - x^k)^6. - Vaclav Kotesovec, Aug 02 2025

A282433 Coefficients in q-expansion of E_6^5, where E_6 is the Eisenstein series A013973.

Original entry on oeis.org

1, -2520, 2457000, -1113204960, 199879986600, 4992350445936, -3054519828108000, -316433406335739840, -15444821445342229080, -469944493113793897080, -9973874479528786860432, -158211337782226162119840, -1972932224893221543809760

Offset: 0

Seiichi Manyama, Feb 15 2017

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

Cf. A282431 (E_2^5), A282015 (E_4^5), this sequence (E_6^5).

Cf. A013973 (E_6), A280869 (E_6^2), A282253 (E_6^3), A282331 (E_6^4), this sequence (E_6^5).

terms = 13;
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E6[x]^5 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A308285 Coefficients in q-expansion of E_2^6, where E_2 is the Eisenstein series A006352.

Original entry on oeis.org

1, -144, 8208, -225216, 2634192, 1488672, -209742912, -503961984, 8575185744, 91347182640, 524570699232, 2230073940672, 7794083954880, 23627036677536, 64145226215808, 159373702203264, 368012313906768, 798872890993632, 1644874069475664, 3234829827767616

Offset: 0

Seiichi Manyama, May 18 2019

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

E_2^b: A006352 (b=1), A281374 (b=2), A282018 (b=3), A282210 (b=4), A282431 (b=5), this sequence (b=6).

cp's OEIS Frontend

A280022 Expansion of phi_{5, 4}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

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A282433 Coefficients in q-expansion of E_6^5, where E_6 is the Eisenstein series A013973.

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A308285 Coefficients in q-expansion of E_2^6, where E_2 is the Eisenstein series A006352.

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