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 1-9 of 9 results.

A029830 Eisenstein series E_20(q) (alternate convention E_10(q)), multiplied by 174611.

Original entry on oeis.org

174611, 13200, 6920614800, 15341851377600, 3628395292275600, 251770019531263200, 8043563916910526400, 150465416446925500800, 1902324110996589786000, 17831242688625346952400, 132000251770026451864800, 807299993919072011054400, 4217144038884527916580800, 19297347832955888660949600
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • J.-P. Serre, Course in Arithmetic, Chap. VII, Section 4.

Links

Crossrefs

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A008410 (E_8), A013974 (E_10), A029828 (691*E_12), A058550 (E_14), A029829 (3617*E_16), A279892 (43867*E_18), this sequence (174611*E_20), A279893 (77683*E_22), A029831 (236364091*E_24).
Cf. A282015 (E_4^5), A282292 (E_4^2*E_6^2 = E_10^2).

Programs

  • Mathematica
    terms = 14;
E20[x_] = 174611 + 13200*Sum[k^19*x^k/(1 - x^k), {k, 1, terms}];
E20[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)
  • PARI
    a(n)=if(n<1,174611*(n==0),13200*sigma(n,19))

Formula

a(n) = 53361*A282015(n) + 121250*A282292(n). - Seiichi Manyama, Feb 11 2017

A282012 Coefficients in q-expansion of E_4^4, where E_4 is the Eisenstein series shown in A004009.

Original entry on oeis.org

1, 960, 354240, 61543680, 4858169280, 137745912960, 2120861041920, 21423820362240, 158753769048000, 928983317334720, 4512174992346240, 18847874280625920, 69518972236842240, 230951926208599680, 701949379778818560, 1975788826748167680
Offset: 0

Views

Author

Seiichi Manyama, Feb 04 2017

Keywords

Comments

Also coefficients in q-expansion of E_8^2.

References

  • G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 207.

Links

Crossrefs

Cf. A004009 (E_4), A008410 (E_4^2), A008411 (E_4^3), this sequence (E_4^4), A282015 (E_4^5).
Cf. A281374 (E_2^2), A008410 (E_4^2), A280869 (E_6^2), this sequence (E_8^2), A282292 (E_10^2).
Cf. A013963, A027364, A281876.

Programs

  • Mathematica
    terms = 16;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E4[x]^4 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Formula

G.f.: (1 + 240 Sum_{i>=1} i^3 q^i/(1-q^i))^4.
16320 * A013963(n) = 3617 * a(n) - 3456000 * A027364(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.

A282431 Coefficients in q-expansion of E_2^5, where E_2 is the Eisenstein series A006352.

Original entry on oeis.org

1, -120, 5400, -104160, 511800, 6770736, -19504800, -452207040, -2959622280, -12932941080, -44497080432, -129918587040, -335811977760, -788655411600, -1714912983360, -3498061536576, -6761506680840, -12481939678320, -22138262633160, -37922739116640
Offset: 0

Views

Author

Seiichi Manyama, Feb 15 2017

Keywords

Links

Crossrefs

Cf. this sequence (E_2^5), A282015 (E_4^5), A282433 (E_6^5).
Cf. A006352 (E_2), A281374 (E_2^2), A282018 (E_2^3), A282210 (E_2^4), this sequence (E_2^5).

Programs

  • Mathematica
    terms = 20;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E2[x]^5 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A282402 Coefficients in q-expansion of E_4^7, where E_4 is the Eisenstein series A004009.

Original entry on oeis.org

1, 1680, 1224720, 505659840, 129351117840, 21060890131680, 2160822606183360, 134717272385473920, 4957295423282269200, 119288258695393463760, 2051465861242156554720, 26894077218337493424960, 281803532524538902825920
Offset: 0

Views

Author

Seiichi Manyama, Feb 14 2017

Keywords

Links

Crossrefs

Cf. A004009 (E_4), A008410 (E_4^2), A008411 (E_4^3), A282012 (E_4^4), A282015 (E_4^5), A282330 (E_4^6), this sequence (E_4^7).

Programs

  • Mathematica
    terms = 13;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E4[x]^7 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

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

Views

Author

Seiichi Manyama, Feb 15 2017

Keywords

Links

Crossrefs

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).

Programs

  • Mathematica
    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 *)

A282474 Coefficients in q-expansion of E_4^8, where E_4 is the Eisenstein series A004009.

Original entry on oeis.org

1, 1920, 1630080, 803228160, 253366181760, 53205643249920, 7498254194403840, 699684356363412480, 42100628403784982400, 1614922125605880493440, 42332208491309728078080, 812648422343847344279040, 12060223533365891970132480
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2017

Keywords

Links

Crossrefs

Cf. A004009 (E_4), A008410 (E_4^2), A008411 (E_4^3), A282012 (E_4^4), A282015 (E_4^5), A282330 (E_4^6), A282402 (E_4^7), this sequence (E_4^8).

Programs

  • Mathematica
    terms = 13;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E4[x]^8 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A299955 Coefficients in expansion of E_4^(3/2).

Original entry on oeis.org

1, 360, 24840, -465120, 57417480, -6800282640, 930889890720, -139401582644160, 22250341370421000, -3723955494287559480, 646515765251485521840, -115559140273640812421280, 21150946022800731753255840, -3948247836773858791840263120
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2018

Keywords

Links

Crossrefs

E_4^(k/8): A108091 (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5), A289318 (k=6), A289319 (k=7), A004009 (k=8), this sequence (k=12), A008410 (k=16), A008411 (k=24), A282012 (k=32), A282015 (k=40).

Formula

Convolution cube of A289292.
a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / n^(5/2), where c = 81*Gamma(1/3)^27 / (32768*sqrt(2)*Pi^(37/2)) = 0.39832876770813443250501819621900549862424768734... - Vaclav Kotesovec, Mar 05 2018

A004671 Theta series of extremal even unimodular lattice in dimension 40.

Original entry on oeis.org

1, 0, 39600, 87859200, 20779902000, 1441891123200, 46065617928000, 861717856665600, 10894640750334000, 102119813013504000, 755967560945968800, 4623420024622080000, 24151651608982497600, 110516220812493619200
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			G.f.: 1 + 39600*q^2 + 87859200*q^3 + ...

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 195.

Links

Programs

  • Sage
    e4 = eisenstein_series_qexp(4,20,normalization = "integral");
delta = CuspForms(1,12).0.q_expansion(20);
e4^5-1200*e4^2*delta  # Andy Huchala, May 07 2021

Formula

a(n) = A282015(n) - 1200 * A037945(n) - Andy Huchala, May 07 2021
Showing 1-9 of 9 results.

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

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