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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A282154 Coefficients in expansion of Eisenstein series -q*(d/dq)(q*(d/dq)E_2).

Original entry on oeis.org

0, 24, 288, 864, 2688, 3600, 10368, 9408, 23040, 25272, 43200, 34848, 96768, 56784, 112896, 129600, 190464, 124848, 303264, 173280, 403200, 338688, 418176, 304704, 829440, 465000, 681408, 699840, 1053696, 605520, 1555200, 738048, 1548288, 1254528, 1498176
Offset: 0

Views

Author

Seiichi Manyama, Feb 07 2017

Keywords

Links

Crossrefs

Cf. A006352 (E_2), A076835 (-q*(d/dq)E_2), this sequence (-q*(d/dq)(q*(d/dq)E_2)).
Cf. A013973 (E_6), A282018 (E_2^3), A282019 (E_2*E_4), A282097.
This sequence is related to A126858.

Programs

  • Mathematica
    terms = 35;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
-x*D[x*D[E2[x], x], x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

Formula

-q*(d/dq)(q*(d/dq)E_2) = -q*(d/dq)((E_2^2 - E_4)/12) = -(E_2^3 - 3*E_2*E_4 + 2*E_6)/72.
a(n) = -(A282018(n) - 3*A282019(n) + 2*A013973(n))/72.
a(n) = 24*A282097(n).

A282020 Coefficients in q-expansion of (E_2^3 - E_2*E_4)/288, where E_2 and E_4 are the Eisenstein series shown in A006352 and A004009, respectively.

Original entry on oeis.org

0, -1, 18, 204, 788, 2250, 4968, 9688, 17640, 27747, 45900, 64548, 98448, 128674, 188496, 232200, 326864, 386478, 537354, 608380, 819000, 926688, 1214136, 1323144, 1758240, 1852625, 2401308, 2584440, 3252256, 3385170, 4374000, 4433248, 5604768, 5840208, 7143876, 7232400, 9239364, 9058858
Offset: 0

Views

Author

N. J. A. Sloane, Feb 06 2017

Keywords

Links

Crossrefs

Cf. A282018 (E_2^3), A282019 (E_2*E_4).

Programs

  • Maple
    with(numtheory); M:=100;
E := proc(k) local n, t1; global M;
t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n, n=1..M+1);
series(t1, q, M+1); end;
e2:=E(2); e4:=E(4); e6:=E(6);
series((e2^3-e2*e4)/288,q,M+1);
seriestolist(%);
  • Mathematica
    terms = 38;
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 + 1}];
(E2[x]^3 - E2[x]*E4[x])/288 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

Formula

a(n) = (A282018(n) - A282019(n))/288. - Seiichi Manyama, Feb 06 2017

A341842 Coefficients of the series whose 12th power equals E_2*E_4, where E_2 and E_4 are the Eisenstein series shown in A006352 and A004009.

Original entry on oeis.org

1, 18, -2088, 301296, -50784174, 9174627360, -1734603719472, 338286925650240, -67486440186470016, 13697820033167444178, -2818359890320927630320, 586296297186462310481424, -123077156275866375661524864, 26034142700316716015964656544
Offset: 0

Views

Author

Peter Bala, Feb 21 2021

Keywords

Comments

The g.f. is the 12th root of the g.f. of A282019.
It is easy to see that E_2(x)*E_4(x) == 1 - 24*Sum_{k >= 1} (k - 10*k^3)*x^k/(1 - x^k) (mod 72), and also that the integer k - 10*k^3 is always divisible by 3. Hence, E_2(x)*E_4(x) == 1 (mod 72). It follows from Heninger et al., p. 3, Corollary 2, that the series expansion of (E_2(x)*E_4(x))^(1/12) = 1 + 18*x - 2088*x^2 + 301296*x^3 - 50784174*x^4 + ... has integer coefficients.

Links

Crossrefs

A004009, A006352, A108091, A282019, A289392.

Programs

  • Maple
    E(2,x) := 1 -  24*add(k*x^k/(1-x^k),   k = 1..20):
E(4,x) := 1 + 240*add(k^3*x^k/(1-x^k), k = 1..20):
with(gfun): series((E(2,x)*E(4,x))^(1/12), x, 20):
seriestolist(%);
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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