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 1-4 of 4 results.

A029831 Eisenstein series E_24(q) (alternate convention E_12(q)), multiplied by 236364091.

Original entry on oeis.org

236364091, 131040, 1099243323360, 12336522153621120, 9221121336284413920, 1562118530273437631040, 103486260766565509822080, 3586400651444203277717760, 77352372210526124884754400, 1161399411211600265764157280
Offset: 0

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Author

Keywords

References

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

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), A029830 (174611*E_20), A279893 (77683*E_22), this sequence (236364091*E_24).
Cf. A282330 (E_4^6), A282332 (E_4^3*E_6^2), A282331 (E_6^4).

Programs

  • Mathematica
    terms = 10;
    E24[x_] = 236364091 + 131040*Sum[k^23*x^k/(1 - x^k), {k, 1, terms}];
    E24[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)
  • PARI
    a(n)=if(n<1,236364091*(n==0),131040*sigma(n,23))

Formula

a(n) = 49679091*A282330(n) + 176400000*A282332(n) + 10285000*A282331(n). - Seiichi Manyama, Feb 12 2017

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

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Author

Seiichi Manyama, Feb 14 2017

Keywords

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 *)

A010839 Expansion of Product_{k >= 1} (1-x^k)^48.

Original entry on oeis.org

1, -48, 1080, -15040, 143820, -985824, 4857920, -16295040, 28412910, 38671600, -424520544, 1268350272, -1211937160, -4306546080, 18293091840, -23522231424, -26299018683, 137218594320, -150999182320, -134713340160
Offset: 0

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Keywords

Examples

			1 - 48*x + 1080*x^2 - 15040*x^3 + 143820*x^4 - 985824*x^5 + 4857920*x^6 - 16295040*x^7 + ...
		

References

  • Morris Newman, A table of the coefficients of the powers of eta(tau), Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.

Crossrefs

Column k=48 of A286354.
Cf. A000203, A082558, A126581, A282330 (E_8^3), A282332 (E_6*E_8*E_10 = E4*E_10^2), A290009, A290010.

Formula

Let b(q) be the determinant of the 3 X 3 Hankel matrix [E_4, E_6, E_8 ; E_6, E_8, E_10 ; E_8, E_10, E_12]. G.f. is -691*b(q)/(q^2*1728^2*250). - Seiichi Manyama, Jul 17 2017
a(n) = (A290010(n+2) - A290009(n+2) + 691*(A282330(n+2) - A282332(n+2)))/(1728^2*250). - Seiichi Manyama, Jul 19 2017
a(0) = 1, a(n) = -(48/n) * Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Aug 13 2023

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

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 *)
Showing 1-4 of 4 results.