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 1-10 of 13 results. Next

A282102 Coefficients in q-expansion of E_2*E_4*E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

1, -288, -129168, -1927296, 65152656, 1535768640, 15223408704, 98001292032, 474055120080, 1870878793824, 6312358836000, 18835985199744, 50831420617152, 126257508465984, 292348744636032, 637474437331200, 1319883180896592, 2610964045674432, 4963491913583664
Offset: 0

Views

Author

Seiichi Manyama, Feb 06 2017

Keywords

Comments

The series expansion of the 12th root of the generating function gives A341801. - Peter Bala, Feb 23 2021

Links

Crossrefs

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A013974 (E_10).
Cf. A281374 (E_2^2), A282019 (E_2*E_4), A282096 (E_2*E_6), A282101 (E_2*E_8), this sequence (E_2*E_10), A341801.

Programs

  • Mathematica
    terms = 19;
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}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E2[x]*E4[x]*E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

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)

A282101 Coefficients in q-expansion of E_2*E_4^2, where E_2, E_4 are the Eisenstein series shown in A006352, A004009, respectively.

Original entry on oeis.org

1, 456, 50328, -470496, -21784008, -234371664, -1446514848, -6502690752, -23328111240, -71276388312, -191952331632, -468159788448, -1052750026272, -2212261706256, -4394299104576, -8303419066176, -15060718806024, -26284654025712, -44471780630856
Offset: 0

Views

Author

Seiichi Manyama, Feb 06 2017

Keywords

Links

Crossrefs

Cf. A006352 (E_2), A004009 (E_4), A008410 (E_8).
Cf. A281374 (E_2^2), A282019 (E_2*E_4), A282096 (E_2*E_6), this sequence (E_2*E_8), A282102 (E_2*E_10).

Programs

  • Mathematica
    terms = 19;
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]*E4[x]^2 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

A282096 Coefficients in q-expansion of E_2*E_6, where E_2, E_6 are the Eisenstein series shown in A006352, A013973, respectively.

Original entry on oeis.org

1, -528, -4608, 312384, 3664416, 21745440, 86782464, 276703872, 741794400, 1758969264, 3797729280, 7568097984, 14222957952, 25253852064, 43166426112, 70518360960, 112406614752, 172631876832, 260795119104, 381636168000, 552633117120, 778105665024
Offset: 0

Views

Author

Seiichi Manyama, Feb 06 2017

Keywords

Links

Crossrefs

Cf. A006352 (E_2), A013973 (E_6).
Cf. A281374 (E_2^2), A282019 (E_2*E_4), this sequence (E_2*E_6), A282101 (E_2*E_8), A282102 (E_2*E_10).

Programs

  • Mathematica
    terms = 22;
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]*E6[x] + 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 *)

A282549 Coefficients in q-expansion of E_2*E_4^3, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

Original entry on oeis.org

1, 696, 161928, 12599904, -22912728, -6132581424, -107015308128, -1012991092032, -6676225539480, -34225591158312, -145164618698832, -530958452207328, -1722320395791072, -5059903726594416, -13673185634909376, -34406198518205376, -81397333990275864
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2017

Keywords

Links

Crossrefs

Cf. A282019 (E_2*E_4), A282101 (E_2*E_4^2), this sequence (E_2*E_4^3), A282546 (E_2*E_4^4).

Programs

  • Mathematica
    terms = 17;
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]*E4[x]^3 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A282752 Coefficients in q-expansion of E_2^2*E_4^2, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

Original entry on oeis.org

1, 432, 39312, -1711296, -14159664, 317412000, 5783500224, 47251354752, 263098098000, 1138294453104, 4105673192160, 12882680040384, 36171259008192, 92764213434144, 220523509245312, 491705284878720, 1037366470830672, 2086141009345632, 4022101701933264
Offset: 0

Views

Author

Seiichi Manyama, Feb 21 2017

Keywords

Links

Crossrefs

Cf. A282019 (E_2*E_4), A282208 (E_2^2*E_4), A282101 (E_2*E_4^2).

Programs

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

A282586 Coefficients in q-expansion of E_2^3*E_4, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

Original entry on oeis.org

1, 168, -13608, 210336, 1805496, -22562064, -322437024, -2063087808, -9165872520, -32250917496, -96383477232, -254377990944, -608736541728, -1346209592784, -2786771573568, -5459635814976, -10197462567432, -18283324047408, -31620880746504
Offset: 0

Views

Author

Seiichi Manyama, Feb 19 2017

Keywords

Links

Crossrefs

Cf. A282019 (E_2*E_4), A282208 (E_2^2*E_4), this sequence (E_2^3*E_4).

Programs

  • Mathematica
    terms = 19;
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]^3*E4[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A280024 Coefficients in q-expansion of E_2^4*E_4, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

Original entry on oeis.org

1, 144, -17712, 524736, -2279088, -79760160, 71126208, 7093116288, 65399933520, 370698709968, 1592500629600, 5659924638528, 17465468914368, 48233085519456, 121766302456704, 285303917520000, 627654170451024, 1308136029869088, 2601247015228176
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2017

Keywords

Links

Crossrefs

Cf. A282019 (E_2*E_4), A282208 (E_2^2*E_4), A282586 (E_2^3*E_4), this sequence (E_2^4*E_4).

Programs

  • Mathematica
    terms = 19;
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]^4*E4[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A282546 Coefficients in q-expansion of E_2*E_4^4, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

Original entry on oeis.org

1, 936, 331128, 52972704, 3355523352, 16684536816, -1540796901408, -39871325253312, -522168659242920, -4651083548616312, -31647933913392432, -175516717881381408, -827283695234707872, -3413277291552455376, -12598120840018061376, -42296015537631706176
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2017

Keywords

Links

Crossrefs

Cf. A006352 (E_2), A004009 (E_4), A282012 (E_4^4).
Cf. A282019 (E_2*E_4), A282101 (E_2*E_4^2), A282549 (E_2*E_4^3), this sequence (E_2*E_4^4).

Programs

  • Mathematica
    terms = 16;
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]* E4[x]^4 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)
Showing 1-10 of 13 results. Next

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

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