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.

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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 *)

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

Views

Author

Seiichi Manyama, Feb 05 2017

Keywords

References

  • G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 208.

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)

Formula

-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

Views

Author

Seiichi Manyama, Feb 05 2017

Keywords

References

  • G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 208.

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)

Formula

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

A282330 Coefficients in q-expansion of E_4^6, where E_4 is the Eisenstein series A004009.

Original entry on oeis.org

1, 1440, 876960, 292072320, 57349833120, 6660135541440, 436536302762880, 15172132360815360, 327295477379498400, 4913576699608450080, 55439481453769056960, 496426192564963006080, 3672749219557161663360, 23148323907214334109120
Offset: 0

Views

Author

Seiichi Manyama, Feb 12 2017

Keywords

Links

Crossrefs

Cf. A004009 (E_4), A008410 (E_4^2), A008411 (E_4^3), A282012 (E_4^4), A282015 (E_4^5), this sequence (E_4^6).

Programs

  • Mathematica
    terms = 14;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E4[x]^6 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

Formula

G.f.: (1 + 240 Sum_{i>=1} i^3 q^i/(1-q^i))^6.

A282332 Coefficients in q-expansion of E_4^3*E_6^2, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Original entry on oeis.org

1, -288, -325728, 11700864, 35176468896, 6601058210880, 438061091013504, 15173572442740992, 327251435243536800, 4913611331706352224, 55439979246339307200, 496425441863436557184, 3672747479405396310912, 23148319784349233726784
Offset: 0

Views

Author

Seiichi Manyama, Feb 12 2017

Keywords

Links

Crossrefs

Cf. A280869 (E_6^2), A282287 (E_4*E_6^2), A282292 (E_4^2*E_6^2 = E_10^2), this sequence (E_4^3*E_6^2).

Programs

  • Mathematica
    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]^3*E6[x]^2 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

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.

Original entry on oeis.org

1, -48, -196128, -33542976, -678319104, 12136422240, 509314518144, 7469015889792, 68272650653760, 458377820557584, 2454769903187520, 11035857376010304, 43103740076823552, 149954656815201504, 473331019057949952, 1375248429330791040, 3719662610125117632
Offset: 0

Views

Author

Seiichi Manyama, Feb 19 2017

Keywords

Links

Crossrefs

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

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}];
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 *)

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 *)

A282547 Coefficients in q-expansion of E_2*E_4*E_6^2, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

1, -792, -648, 67840416, 3219716376, 16790031216, -1536150710304, -39898324202688, -522122582192040, -4650999065751096, -31648313780323632, -175516685804469024, -827282698744164768, -3413275186936731984, -12598131165680789568, -42296014044574387776
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2017

Keywords

Links

Crossrefs

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6).

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}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E2[x]*E4[x]*E6[x]^2 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A282586 Coefficients in q-expansion of E_2^3*E_4, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

Original entry on oeis.org

1, 168, -13608, 210336, 1805496, -22562064, -322437024, -2063087808, -9165872520, -32250917496, -96383477232, -254377990944, -608736541728, -1346209592784, -2786771573568, -5459635814976, -10197462567432, -18283324047408, -31620880746504
Offset: 0

Views

Author

Seiichi Manyama, Feb 19 2017

Keywords

Links

Crossrefs

Cf. A282019 (E_2*E_4), A282208 (E_2^2*E_4), this sequence (E_2^3*E_4).

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]^3*E4[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)
Previous Showing 21-30 of 196 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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