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 1-5 of 5 results.

A290048 Coefficients in expansion of E_6*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

Original entry on oeis.org

1, -552, 8640, 116000, -4868460, 67855536, -544522240, 2742137280, -8237774250, 10592091400, 3366617856, 113971542048, -1217020425880, 4535746506000, -5415752171520, -19090509870144, 93580817811453, -142801363479240, -80721277168000, 665065363025280
Offset: 2

Views

Author

Seiichi Manyama, Jul 19 2017

Keywords

Links

Crossrefs

Cf. A000594, A010839, A013973 (E_6).
Cf. A282382, A282461 (E_6*E_10*E_14 = E_10^3), A290049, A290050.
E_k*Delta^2: A290178 (k=4), this sequence (k=6), A290180 (k=8), A290181 (k=10), A290182 (k=14).

Programs

  • Mathematica
    terms = 20;
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E6[x]*QPochhammer[x]^48 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Formula

Let b(q) be the determinant of the 3 X 3 Hankel matrix [E_6, E_8, E_10 ; E_8, E_10, E_12 ; E_10, E_12, E_14]. G.f. is -691^2*b(q)/(1728^2*250^2).
a(n) = (A290050(n) - 2*691*A290049(n) + 691^2*A282382(n))/(1728^2*250^2).

A290178 Coefficients in expansion of E_4*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

Original entry on oeis.org

1, 192, -8280, 147200, -1438020, 7491456, -4626880, -246965760, 2112385950, -9443825600, 23625035616, -14413771008, -118710609640, 427914230400, -467038103040, -645319017984, 1640006523477, 2800373100480, -8506579320400, -21655683517440, 108181106829972
Offset: 2

Views

Author

Seiichi Manyama, Jul 23 2017

Keywords

Links

Crossrefs

E_k*Delta^2: this sequence (k=4), A290048 (k=6), A290180 (k=8), A290181 (k=10), A290182 (k=14).
Cf. A000594, A004009 (E_4), A290152.

Programs

  • Mathematica
    terms = 21;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E4[x]*QPochhammer[x]^48 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Formula

Let b(q) be the determinant of the 3 X 3 matrix [E_4, E_8, E_10 ; E_6, E_10, E_12 ; E_8, E_12, E_14]. G.f. is 691^2*b(q)/(1728^2*21^2*250).

A037946 Coefficients of unique normalized cusp form Delta_22 of weight 22 for full modular group.

Original entry on oeis.org

1, -288, -128844, -2014208, 21640950, 37107072, -768078808, 1184071680, 6140423133, -6232593600, -94724929188, 259518615552, -80621789794, 221206696704, -2788306561800, 3883087691776, 3052282930002
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			q^2 - 288*q^4 - ...

References

  • G. Harder. "A Congruence Between a Siegel and an Elliptic Modular Form." The 1-2-3 of modular forms. Springer Berlin Heidelberg, 2008. 247-262.

Links

Crossrefs

Cf. A000594 ((E_4(q)^3 - E_6(q)^2)/12^3), A004009 (E_4(q)), A013969, A013973 (E_6(q)), A290181.

Programs

  • Mathematica
    terms = 17;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms+1}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms+1}];
((E4[x]^3 - E6[x]^2)/12^3)*E4[x]*E6[x] + O[x]^(terms+1) // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Feb 27 2018, after Seiichi Manyama *)

Formula

a(n) == A013969(n) mod 77683. - Seiichi Manyama, Feb 03 2017
G.f.: (E_4(q)^3 - E_6(q)^2)/12^3 * E_4(q) * E_6(q). - Seiichi Manyama, Jun 09 2017
G.f.: 691/(1728*250) * (E_8(q)*E_14(q) - E_10(q)*E_12(q)). - Seiichi Manyama, Jul 25 2017

A290180 Coefficients in expansion of E_8*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

Original entry on oeis.org

1, 432, 39960, -1418560, 17312940, -71928864, -462815680, 7500885120, -38038437810, 29000909200, 729783353376, -4661016429888, 13691625085880, -16503845217120, -14982974507520, 45085348093056, 99234456545637, -157805792764560, -1644659689877680
Offset: 2

Views

Author

Seiichi Manyama, Jul 23 2017

Keywords

Links

Crossrefs

E_k*Delta^2: A290178 (k=4), A290048 (k=6), this sequence (k=8), A290181 (k=10), A290182 (k=14).
Cf. A000594, A008410 (E_8).

Programs

  • Mathematica
    terms = 19;
E8[x_] = 1 + 480*Sum[k^7*x^k/(1 - x^k), {k, 1, terms}];
E8[x]*QPochhammer[x]^48 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Formula

Let b(q) be the determinant of the 3 X 3 matrix [E_6, E_8, E_12 ; E_8, E_10, E_14 ; E_10, E_12, E_16]. G.f. is -691^2*3617*b(q)/(1728^2*2^3*3*5^3*7^2*467).

A290182 Coefficients in expansion of E_14*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

Original entry on oeis.org

1, -72, -194400, -28866400, 13994100, 9650004336, -99683138560, -1007380800, 5570606272950, -32186306471000, -2717893793664, 724443400725408, -2662202398202200, -401005712372400, 19385312101171200, 24633489938571456, -449375771787124707
Offset: 2

Views

Author

Seiichi Manyama, Jul 23 2017

Keywords

Links

Crossrefs

E_k*Delta^2: A290178 (k=4), A290048 (k=6), A290180 (k=8), A290181 (k=10), this sequence (k=14).
Cf. A000594, A058550 (E_14).

Programs

  • Mathematica
    terms = 17;
E14[x_] = 1 - 24*Sum[k^13*x^k/(1 - x^k), {k, 1, terms}];
E14[x]*QPochhammer[x]^48 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Formula

Let b(q) be the determinant of the 3 X 3 matrix [E_8, E_10, E_14 ; E_10, E_12, E_16 ; E_12, E_14, E_18]. G.f. is -691^2*3617*43867*b(q)/(1728^2*2^6*3*5^3*7^2*97*7213).
Showing 1-5 of 5 results.

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

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