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.

A282097 Coefficients in q-expansion of (3*E_2*E_4 - 2*E_6 - E_2^3)/1728, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 12, 36, 112, 150, 432, 392, 960, 1053, 1800, 1452, 4032, 2366, 4704, 5400, 7936, 5202, 12636, 7220, 16800, 14112, 17424, 12696, 34560, 19375, 28392, 29160, 43904, 25230, 64800, 30752, 64512, 52272, 62424, 58800, 117936, 52022, 86640, 85176, 144000, 70602
Offset: 0

Views

Author

Seiichi Manyama, Feb 06 2017

Keywords

Comments

Multiplicative because A000203 is. - Andrew Howroyd, Jul 25 2018

Examples

			a(6) = 1^3*6^2 + 2^3*3^2 + 3^3*2^2 + 6^3*1^2 = 432.

Links

Crossrefs

Cf. this sequence (phi_{3, 2}), A282099 (phi_{5, 2}).
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282018 (E_2^3), A282019 (E_2*E_4).
Cf. A000203 (sigma(n)), A064987 (n*sigma(n)), this sequence (n^2*sigma(n)), A282211 (n^3*sigma(n)).
Cf. A222171.

Programs

  • Magma
    [0] cat [n^2*DivisorSigma(1, n): n in [1..50]]; // Vincenzo Librandi, Mar 01 2018
  • Mathematica
    a[0]=0;a[n_]:=(n^2)*DivisorSigma[1,n];Table[a[n],{n,0,41}] (* Indranil Ghosh, Feb 21 2017 *)
terms = 42; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[(3*Ei[2]*Ei[4] - 2*Ei[6] - Ei[2]^3)/1728 + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)
  • PARI
    a(n) = if (n==0, 0, n^2*sigma(n)); \\ Michel Marcus, Feb 21 2017

Formula

a(n) = (3*A282019(n) - 2*A013973(n) - A282018(n))/1728.
G.f.: phi_{3, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
a(n) = n^2*A000203(n) for n > 0. - Seiichi Manyama, Feb 19 2017
G.f.: Sum_{k>=1} k^3*x^k*(1 + x^k)/(1 - x^k)^3. - Ilya Gutkovskiy, May 02 2018
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^(2*e) * (p^(e+1)-1)/(p-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-3).
Sum_{k=1..n} a(k) ~ (Pi^2/24) * n^4. (End)
From Peter Bala, Jan 22 2024: (Start)
a(n) = Sum_{1 <= i, j, k <= n} sigma_2( gcd(i, j, k, n) ).
a(n) = Sum_{1 <= i, j <= n} sigma_3( gcd(i, j, n) ).
a(n) = Sum_{d divides n} sigma_2(d) * J_3(n/d) = Sum_{d divides n} sigma_3(d) * J_2(n/d), where the Jordan totient functions J_2(n) = A007434(n) and J_3(n) = A059376(n). (End)

A282253 Coefficients in q-expansion of E_6^3, where E_6 is the Eisenstein series shown in A013973.

Original entry on oeis.org

1, -1512, 712152, -78097824, -11474230824, -498089967984, -11088580243104, -152351956669248, -1474676091461160, -10921529499813576, -65490182325115632, -331010378444247264, -1452953351890984608, -5665062963045803184, -19968586384352171328
Offset: 0

Views

Author

Seiichi Manyama, Feb 10 2017

Keywords

Links

Crossrefs

Cf. A282018 (E_2^3), A008411 (E_4^3), this sequence (E_6^3).
Cf. A013973 (E_6), A280869 (E_6^2), this sequence (E_6^3).

Programs

  • Mathematica
    terms = 15;
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E6[x]^3 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

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 *)

A282210 Coefficients in q-expansion of E_2^4, where E_2 is the Eisenstein series shown in A006352.

Original entry on oeis.org

1, -96, 3168, -34944, -107808, 1955520, 16829568, 76708608, 258593760, 715480608, 1729546560, 3771497088, 7581237888, 14296261056, 25520442624, 43590539520, 71582414304, 113752634688, 175604039136, 264097115520, 388619703360, 559658001408, 792716685696
Offset: 0

Views

Author

Seiichi Manyama, Feb 09 2017

Keywords

Links

Crossrefs

Cf. A006352 (E_2), A281374 (E_2^2), A282018 (E_2^3), this sequence (E_2^4).

Programs

  • Mathematica
    terms = 23;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E2[x]^4 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A282595 Coefficients in q-expansion of E_2^2*E_6, where E_2 and E_6 are respectively the Eisenstein series A006352 and A013973.

Original entry on oeis.org

1, -552, 7992, 460896, -3450504, -88161264, -728085024, -3775195968, -14894175240, -48567693576, -137214605232, -347495426784, -804758753568, -1733365307184, -3511286411328, -6753825302976, -12422812497672, -21971174382288, -37567247938344
Offset: 0

Views

Author

Seiichi Manyama, Feb 19 2017

Keywords

Links

Crossrefs

Cf. A282018 (E_2^3), this sequence (E_2^2*E_6), A282576 (E_2*E_6^2), A282253 (E_6^3).

Programs

  • Mathematica
    terms = 19;
E2[x_] = 1 - 24*Sum[k*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]^2*E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

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 *)

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).

A308285 Coefficients in q-expansion of E_2^6, where E_2 is the Eisenstein series A006352.

Original entry on oeis.org

1, -144, 8208, -225216, 2634192, 1488672, -209742912, -503961984, 8575185744, 91347182640, 524570699232, 2230073940672, 7794083954880, 23627036677536, 64145226215808, 159373702203264, 368012313906768, 798872890993632, 1644874069475664, 3234829827767616
Offset: 0

Views

Author

Seiichi Manyama, May 18 2019

Keywords

Links

Crossrefs

E_2^b: A006352 (b=1), A281374 (b=2), A282018 (b=3), A282210 (b=4), A282431 (b=5), this sequence (b=6).

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
Showing 1-9 of 9 results.

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

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