cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A282060 Coefficients in q-expansion of E_4*(E_2*E_4 - E_6)/720, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 258, 6564, 66052, 390630, 1693512, 5764808, 16909320, 43066413, 100782540, 214358892, 433565328, 815730734, 1487320464, 2564095320, 4328785936, 6975757458, 11111134554, 16983563060, 25801892760, 37840199712, 55304594136, 78310985304, 110992776480
Offset: 0

Views

Author

Seiichi Manyama, Feb 05 2017

Keywords

Comments

Multiplicative because A013955 is. - Andrew Howroyd, Jul 25 2018

Examples

			a(6) = 1^8*6^1 + 2^8*3^1 + 3^8*2^1 + 6^8*1^1 = 1693512.

Links

Crossrefs

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), A282050 (phi_{6, 1}), this sequence (phi_{8, 1}).
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282101 (E_2*E_4^2), A013974 (E_4*E_6 = E_10).
Cf. A013955, A013666, A196288.

Programs

  • Mathematica
    terms = 25;
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]*(E2[x]*E4[x] - E6[x])/720 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)
Table[n * DivisorSigma[7, n], {n, 0, 24}] (* Amiram Eldar, Sep 06 2023 *)
nmax = 40; CoefficientList[Series[x*Sum[k^8*x^(k-1)/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 01 2025 *)
  • PARI
    a(n) = if(n < 1, 0, n*sigma(n, 7)) \\ Andrew Howroyd, Jul 25 2018

Formula

G.f.: phi_{8, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
a(n) = (A282101(n) - A013974(n))/720. - Seiichi Manyama, Feb 10 2017
If p is a prime, a(p) = p^8 + p = A196288(p). - Seiichi Manyama, Feb 10 2017
a(n) = n*A013955(n) for n > 0. - Seiichi Manyama, Feb 18 2017
Sum_{k=1..n} a(k) ~ zeta(8) * n^9 / 9. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(7*e+7)-1)/(p^7-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-8). (End)
G.f. Sum_{k>=1} k^8*x^(k-1)/(1 - x^k)^2. - Vaclav Kotesovec, Aug 01 2025

A282102 Coefficients in q-expansion of E_2*E_4*E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

1, -288, -129168, -1927296, 65152656, 1535768640, 15223408704, 98001292032, 474055120080, 1870878793824, 6312358836000, 18835985199744, 50831420617152, 126257508465984, 292348744636032, 637474437331200, 1319883180896592, 2610964045674432, 4963491913583664
Offset: 0

Views

Author

Seiichi Manyama, Feb 06 2017

Keywords

Comments

The series expansion of the 12th root of the generating function gives A341801. - Peter Bala, Feb 23 2021

Links

Crossrefs

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A013974 (E_10).
Cf. A281374 (E_2^2), A282019 (E_2*E_4), A282096 (E_2*E_6), A282101 (E_2*E_8), this sequence (E_2*E_10), A341801.

Programs

  • Mathematica
    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}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E2[x]*E4[x]*E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

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

Views

Author

Seiichi Manyama, Feb 22 2017

Keywords

Comments

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

Links

Crossrefs

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.

Programs

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

Formula

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

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

Views

Author

Seiichi Manyama, Feb 06 2017

Keywords

Links

Crossrefs

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).

Programs

  • Mathematica
    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 *)

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

Views

Author

Seiichi Manyama, Feb 09 2017

Keywords

Comments

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

Examples

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

Links

Crossrefs

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.

Programs

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

Formula

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

Views

Author

Seiichi Manyama, Feb 21 2017

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)
  • PARI
    a(n) = if(n < 1, 0, n^2*sigma(n, 5)) \\ Andrew Howroyd, Jul 25 2018

Formula

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

A282208 Coefficients in q-expansion of E_2^2*E_4, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

Original entry on oeis.org

1, 192, -8928, 9984, 1420896, 11433600, 53760384, 187233792, 533725920, 1327018944, 2953851840, 6060858624, 11611915392, 21030301824, 36387585792, 60357358080, 97020376032, 150755202432, 229107724704, 338493223680, 492378465600, 698632525824, 980953593984
Offset: 0

Views

Author

Seiichi Manyama, Feb 09 2017

Keywords

Links

Crossrefs

Cf. A006352 (E_2), A004009 (E_4), A281374 (E_2^2), A282019 (E_2*E_4), A008410 (E_4^2 = E_8), A282018 (E_2^3), this sequence (E_2^2*E_4), A282101 (E_2*E_4^2), A008411 (E_4^3).

Programs

  • Mathematica
    terms = 23;
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]^2*E4[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

A282549 Coefficients in q-expansion of E_2*E_4^3, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

Original entry on oeis.org

1, 696, 161928, 12599904, -22912728, -6132581424, -107015308128, -1012991092032, -6676225539480, -34225591158312, -145164618698832, -530958452207328, -1722320395791072, -5059903726594416, -13673185634909376, -34406198518205376, -81397333990275864
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2017

Keywords

Links

Crossrefs

Cf. A282019 (E_2*E_4), A282101 (E_2*E_4^2), this sequence (E_2*E_4^3), A282546 (E_2*E_4^4).

Programs

  • Mathematica
    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}];
E2[x]*E4[x]^3 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A282752 Coefficients in q-expansion of E_2^2*E_4^2, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

Original entry on oeis.org

1, 432, 39312, -1711296, -14159664, 317412000, 5783500224, 47251354752, 263098098000, 1138294453104, 4105673192160, 12882680040384, 36171259008192, 92764213434144, 220523509245312, 491705284878720, 1037366470830672, 2086141009345632, 4022101701933264
Offset: 0

Views

Author

Seiichi Manyama, Feb 21 2017

Keywords

Links

Crossrefs

Cf. A282019 (E_2*E_4), A282208 (E_2^2*E_4), A282101 (E_2*E_4^2).

Programs

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

A282546 Coefficients in q-expansion of E_2*E_4^4, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

Original entry on oeis.org

1, 936, 331128, 52972704, 3355523352, 16684536816, -1540796901408, -39871325253312, -522168659242920, -4651083548616312, -31647933913392432, -175516717881381408, -827283695234707872, -3413277291552455376, -12598120840018061376, -42296015537631706176
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2017

Keywords

Links

Crossrefs

Cf. A006352 (E_2), A004009 (E_4), A282012 (E_4^4).
Cf. A282019 (E_2*E_4), A282101 (E_2*E_4^2), A282549 (E_2*E_4^3), this sequence (E_2*E_4^4).

Programs

  • Mathematica
    terms = 16;
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]^4 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)
Showing 1-10 of 10 results.

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