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 10 results.

A281372 Coefficients in q-expansion of (E_2*E_4 - E_6)/720, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 18, 84, 292, 630, 1512, 2408, 4680, 6813, 11340, 14652, 24528, 28574, 43344, 52920, 74896, 83538, 122634, 130340, 183960, 202272, 263736, 279864, 393120, 393775, 514332, 551880, 703136, 707310, 952560, 923552, 1198368, 1230768, 1503684, 1517040, 1989396, 1874198, 2346120, 2400216, 2948400
Offset: 0

Views

Author

N. J. A. Sloane, Feb 04 2017

Keywords

Comments

The q-expansion of the square of this expression is given in A281371.
Multiplicative because A001158 is. - Andrew Howroyd, Jul 23 2018

Links

Crossrefs

Cf. A001158, A003840, A006352, A004009, A013973, A064987, A145094, A281371, A328259.

Programs

  • Magma
    [0] cat [n*DivisorSigma(3, n): n in [1..50]]; // Vincenzo Librandi, Mar 01 2018
  • Maple
    with(gfun):
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);
t1:=series((e2*e4-e6)/720,q,M+1);
seriestolist(t1);
# alternative program
seq(add(sigma[4](d)*phi(n/d), d in divisors(n)), n = 1..100); # Peter Bala, Jan 20 2024
  • Mathematica
    Table[If[n==0, 0, n * DivisorSigma[3, n]], {n, 0, 40}] (* Indranil Ghosh, Mar 11 2017 *)
terms = 41; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[(Ei[2] Ei[4] - Ei[6])/720 + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)
  • PARI
    for(n=0, 40, print1(if(n==0, 0, n * sigma(n, 3)), ", ")) \\ Indranil Ghosh, Mar 11 2017

Formula

a(n) = A145094(n)/240 for n > 0. - Seiichi Manyama, Feb 04 2017
G.f.: phi_{4, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}. - Seiichi Manyama, Feb 04 2017
a(n) = n*A001158(n) for n > 0. - Seiichi Manyama, Feb 18 2017
G.f.: x*f'(x), where f(x) = Sum_{k>=1} k^3*x^k/(1 - x^k). - Ilya Gutkovskiy, Aug 31 2017
Sum_{k=1..n} a(k) ~ Pi^4 * n^5 / 450. - Vaclav Kotesovec, May 09 2022
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(3*e+3)-1)/(p^3-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-4). (End)
a(n) = Sum_{k = 1..n} sigma_4( gcd(k, n) ) = Sum_{d divides n} sigma_4(d) * phi(n/d). - Peter Bala, Jan 19 2024
a(n) = Sum_{1 <= i, j, k, l <= n} sigma_1( gcd(i, j, k, l, n) ) = Sum_{d divides n} sigma_1(d) * J_4(n/d), where the Jordan totient function J_4(n) = A059377(n). - Peter Bala, Jan 22 2024

A076835 Coefficients in expansion of Eisenstein series -q*E'_2.

Original entry on oeis.org

24, 144, 288, 672, 720, 1728, 1344, 2880, 2808, 4320, 3168, 8064, 4368, 8064, 8640, 11904, 7344, 16848, 9120, 20160, 16128, 19008, 13248, 34560, 18600, 26208, 25920, 37632, 20880, 51840, 23808, 48384, 38016, 44064, 40320, 78624, 33744, 54720, 52416, 86400
Offset: 1

Views

Author

N. J. A. Sloane, Feb 28 2009

Keywords

Examples

			G.f. = 24*q + 144*q^2 + 288*q^3 + 672*q^4 + 720*q^5 + ...

Links

Crossrefs

Cf. A006352 (E_2), A004009 (E_4), A064987.
Cf. this sequence (-q*E'_2), A145094 (q*E'_4), A145095 (-q*E'_6).

Programs

  • Maple
    with(numtheory); E:=proc(k) series(1-(2*k/bernoulli(k))*add( sigma[k-1](n)*q^n, n=1..60),q,61); end; -diff(E(2),q);
  • Mathematica
    terms = 41;
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])/12 + O[x]^terms // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Feb 23 2018 *)
nmax = 40; Rest[CoefficientList[Series[24*x*Sum[k^2*x^(k-1)/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 01 2025 *)
  • PARI
    a(n) = 24 * n * sigma(n); \\ Amiram Eldar, Jan 07 2025

Formula

q*E'_2 = (E_2^2-E_4)/12.
a(n) = 24*A064987(n).
G.f.: 24*x*f'(x), where f(x) = Sum_{k>=1} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Aug 31 2017

A145095 Coefficients in expansion of Eisenstein series -q*E'_6.

Original entry on oeis.org

504, 33264, 368928, 2130912, 7877520, 24349248, 59298624, 136382400, 268953048, 519916320, 892872288, 1559827584, 2432718288, 3913709184, 5766344640, 8728481664, 12165343344, 17750901168, 23711133600, 33306154560, 43406592768, 58929571008
Offset: 1

Views

Author

N. J. A. Sloane, Feb 28 2009

Keywords

Examples

			G.f. = 504*q + 33264*q^2 + 368928*q^3 + 2130912*q^4 + 7877520*q^5 + ...

Links

Crossrefs

Cf. A076835 (-q*E'_2), A145094 (q*E'_4), this sequence (-q*E'_6).

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}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
-(E2[x]*E6[x] - E4[x]^2)/2 + O[x]^terms // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Feb 23 2018 *)
nmax = 40; Rest[CoefficientList[Series[504*x*Sum[k^6*x^(k-1)/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 01 2025 *)

Formula

q*E'_6 = (E_2*E_6-E_4^2)/2.

A289636 Coefficients in expansion of -q*E'_4/E_4 where E_4 is the Eisenstein Series (A004009).

Original entry on oeis.org

-240, 53280, -12288960, 2835808320, -654403831200, 151013228757120, -34848505552897920, 8041801037378486400, -1855762905734676483120, 428244362959801779806400, -98823634118413525094402880, 22804995243537595828606337280
Offset: 1

Views

Author

Seiichi Manyama, Jul 09 2017

Keywords

Examples

			a(1) = 1 * A110163(1) = -240,
a(2) = 1 * A110163(1) + 2 * A110163(2) = 53280,
a(3) = 1 * A110163(1) + 3 * A110163(3) = -12288960.

Links

Crossrefs

-q*E'_k/E_k: A289635 (k=2), this sequence (k=4), A289637 (k=6), A289638 (k=8), A289639 (k=10), A289640 (k=14).
Cf. A001943, A004009 (E_4), A006352 (E_2), A110163, A145094, A289633.

Programs

  • Mathematica
    nmax = 20; Rest[CoefficientList[Series[-240*x*Sum[k*DivisorSigma[3, k]*x^(k-1), {k, 1, nmax}]/(1 + 240*Sum[DivisorSigma[3, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 09 2017 *)
terms = 12; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[-D[Ei[4], x]/Ei[4] + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)

Formula

a(n) = Sum_{d|n} d * A110163(d) = A289633(n)/6.
a(n) = A288261(n)/3 + 8*A000203(n).
a(n) = -Sum_{k=1..n-1} A004009(k)*a(n-k) - A004009(n)*n.
G.f.: 1/3 * E_6/E_4 - 1/3 * E_2.
a(n) ~ (-1)^n * exp(Pi*sqrt(3)*n). - Vaclav Kotesovec, Jul 09 2017

A126858 Coefficients in quasimodular form F_2(q) of level 1 and weight 6.

Original entry on oeis.org

0, 0, 1, 8, 30, 80, 180, 336, 620, 960, 1590, 2200, 3416, 4368, 6440, 7920, 11160, 13056, 18333, 20520, 27860, 31360, 41052, 44528, 59760, 62400, 80990, 87120, 109872, 113680, 147960, 148800, 188976, 196416, 240210, 243040, 311910, 303696, 376580, 385840
Offset: 0

Views

Author

N. J. A. Sloane, Mar 15 2007

Keywords

Comments

This is also (5*E_2^3 - 3*E_2*E_4 - 2*E_6)/51840, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively. - N. J. A. Sloane, Feb 06 2017
This is also ((q*(d/dq)E_4)/240 + q*(d/dq)(q*(d/dq)E_2)/24)/6, where E_2 and E_4 are the Eisenstein series shown in A006352 and A004009, respectively. - Seiichi Manyama, Feb 08 2017

Examples

			F_2(q) = q^2 + 8*q^3 + 30*q^4 + 80*q^5 + 180*q^6 + 336*q^7 + 620*q^8 + 960*q^9 + 1590*q^10 + 2200*q^11 + ...

Links

Crossrefs

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A145094 (q*(d/dq)E_4), A281372, A282097, A282154 (-q*(d/dq)(q*(d/dq)E_2)).

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((5*e2^3-3*e2*e4-2*e6)/51840,q,M+1);
seriestolist(%); # from N. J. A. Sloane, Feb 06 2017
  • Mathematica
    terms = 40; Ei[n_] = 1 - (2 n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, 1, terms}]; S = 5 Ei[2]^3 - 3 Ei[2] Ei[4] - 2 Ei[6]; CoefficientList[S + O[x]^terms, x]/SeriesCoefficient[S, {x, 0, 2}] (* Jean-François Alcover, Feb 28 2018 *)
  • PARI
    {a(n) = local(L1, L2, L3); if( n<0, 0, L1 = 1 - 24 * sum( k = 1, n, sigma(k) * x^k, x * O(x^n)); L2 = x * L1'; L3 = x * L2'; polcoeff( (L1 * L2 - L3) / 720, n))} /* Michael Somos, Jan 08 2012 */

Formula

F_2(q) = (5*E(2)^3-3*E(2)*E(4)-2*E(6))/51840 where E(k) is the normalized Eisenstein series of weight k (cf. A006352, etc.).
Expansion of (L1 * L2 - L3) / 720 where L1 = E2 (A006352), L2 = q * dL1/dq, L3 = q * dL2/dq in powers of q where E2 is an Eisenstein series. - Michael Somos, Jan 08 2012
a(n) = (A145094(n)/240 - A282154(n)/24)/6 = (A281372(n) - A282097(n))/6. - Seiichi Manyama, Feb 08 2017

A281371 Coefficients in q-expansion of (E_2*E_4 - E_6)^2/518400, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 0, 1, 36, 492, 3608, 18828, 74760, 250352, 717984, 1866558, 4365580, 9635472, 19639032, 38559416, 71222616, 128258496, 219619968, 370366101, 597550068, 955638824, 1471571136, 2253173892, 3335433368, 4932972864, 7064391840, 10133162774, 14128072488, 19743952032, 26864847352
Offset: 0

Views

Author

N. J. A. Sloane, Feb 04 2017

Keywords

Comments

This is (up to a constant factor), the numerator of the expression phi defined in Cohn (2017) (see phi on page 114 of the Notices version).

Links

Crossrefs

Cf. A006352, A004009, A013973, A145094, A281372 (the square root).

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);
t1:=series((e2*e4-e6)^2/518400,q,M+1);
seriestolist(t1);
  • Mathematica
    terms = 30;
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])^2/518400 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A281373 Coefficients in q-expansion of (E_2*E_4 - E_6)^2/(300*(E_6^2-E_4^3)), where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 60, 1680, 30280, 405678, 4369680, 39729200, 315045840, 2230260741, 14340456648, 84870112272, 467160257760, 2411818867430, 11759239565472, 54457051387536, 240692336520352, 1019498573990610, 4152992658207660, 16319887656747248, 62032458633713904, 228608370781579488
Offset: 0

Views

Author

N. J. A. Sloane, Feb 04 2017

Keywords

Comments

This is (up to a constant factor), the function phi defined in Cohn (2017) (see phi on page 114 of the Notices version).

Links

Crossrefs

Cf. A006352, A004009, A013973, A145094, A281371 (the numerator), A000594 (the denominator), A319134, A319294.

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);
t1:=series((e2*e4-e6)^2/518400,q,M+1);
t2:=series((e4^3-e6^2)/1728,q,M+1);
t3:=series(t1/t2,q,M+1);
seriestolist(t3);
  • Mathematica
    terms = 22;
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])^2/(300*(E6[x]^2 - E4[x]^3)) + O[x]^terms // CoefficientList[#, x]& // Abs (* Jean-François Alcover, Feb 27 2018 *)

Formula

a(n) ~ exp(4*Pi*sqrt(n)) / (14400 * sqrt(2) * Pi^2 * n^(7/4)). - Vaclav Kotesovec, Jun 06 2018

A289744 Coefficients in expansion of q*E'_8 where E_8 is the Eisenstein Series (A008410).

Original entry on oeis.org

480, 123840, 3150720, 31704960, 187502400, 812885760, 2767107840, 8116473600, 20671878240, 48375619200, 102892268160, 208111357440, 391550752320, 713913822720, 1230765753600, 2077817249280, 3348363579840, 5333344585920, 8152110268800, 12384908524800
Offset: 1

Views

Author

Seiichi Manyama, Jul 11 2017

Keywords

Links

Crossrefs

(-1)^(k/2)*q*E'_{k}: A076835 (k=2), A145094 (k=4), A145095 (k=6), this sequence (k=8), A289745 (k=10), A289746 (k=14).
Cf. A008410 (E_8), A013955, A282060, A289638.

Programs

Formula

a(n) = 480*A282060(n) = 480*n*A013955(n).

A289745 Coefficients in expansion of -q*E'_10 where E_10 is the Eisenstein Series (A013974).

Original entry on oeis.org

264, 270864, 15589728, 277365792, 2578126320, 15995060928, 74573467584, 284022573120, 920557851048, 2645157604320, 6847480097568, 16379004749184, 36394641851568, 76512377741184, 152243515448640, 290839114879104, 532222389723024, 944492355175248
Offset: 1

Views

Author

Seiichi Manyama, Jul 11 2017

Keywords

Links

Crossrefs

(-1)^(k/2)*q*E'_{k}: A076835 (k=2), A145094 (k=4), A145095 (k=6), A289744 (k=8), this sequence (k=10), A289746 (k=14).
Cf. A013957, A013974 (E_10), A282254, A289639.

Programs

Formula

a(n) = 264*A282254(n) = 264*n*A013957(n).

A289746 Coefficients in expansion of -q*E'_14 where E_14 is the Eisenstein Series (A058550).

Original entry on oeis.org

24, 393264, 114791328, 6443237472, 146484375120, 1880970700608, 16277353748544, 105566002741440, 549043363293048, 2400292970716320, 9113996005998048, 30817824417926784, 94497033256783248, 266720718523641984, 700630664636456640
Offset: 1

Views

Author

Seiichi Manyama, Jul 11 2017

Keywords

Links

Crossrefs

(-1)^(k/2)*q*E'_{k}: A076835 (k=2), A145094 (k=4), A145095 (k=6), A289744 (k=8), A289745 (k=10), this sequence (k=14).
Cf. A013961, A058550 (E_14), A282597, A289640.

Programs

Formula

a(n) = 24*A282597(n) = 24*n*A013961(n).
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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