A145095 -id:A145095 - cp's OEIS frontend

A013973 Expansion of Eisenstein series E_6(q) (alternate convention E_3(q)).

1, -504, -16632, -122976, -532728, -1575504, -4058208, -8471232, -17047800, -29883672, -51991632, -81170208, -129985632, -187132176, -279550656, -384422976, -545530104, -715608432, -986161176, -1247954400, -1665307728, -2066980608, -2678616864, -3243917376, -4159663200

Offset: 0

N. J. A. Sloane

Ramanujan Lambert series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

			G.f. = 1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + ...

W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 53.
R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 53.
M. Kaneko and D. Zagier, Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998.
N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 111.
Jean-Pierre Serre, "A Course in Arithmetic", Springer, 1978
Joseph H. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves", Springer, 1994

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
R. E. Borcherds, Automorphic forms on O_{s+2,2}(R)^{+} and generalized Kac-Moody algebras, pp. 744-752 of Proc. Intern. Congr. Math., Vol. 2, 1994.
D. Bump, Automorphic Forms and Representations, Cambr. Univ. Press, 1997, p. 29.
Heng Huat Chan, Shaun Cooper, and Pee Choon Toh, Ramanujan's Eisenstein series and powers of Dedekind's eta-function, Journal of the London Mathematical Society 75.1 (2007): 225-242. See R(q).
Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
H. Ochiai, Counting functions for branched covers of elliptic curves and quasi-modular forms, arXiv:math-ph/9909023, 1999.
Eric Weisstein's World of Mathematics, Eisenstein Series.
Index entries for sequences related to Eisenstein series

Cf. A004009, A008410, A013974, A145095.
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A008410 (E_8), A013974 (E_10), A029828 (E_12), A058550 (E_14), A029829 (E_16), A029830 (E_20), A029831 (E_24).
Cf. A001160, A286346 (eta(q)^24 / eta(q^2)^12), A286399 (eta(q^2)^12 * eta(q^4)^8 / eta(q)^8).

Basis( ModularForms( Gamma1(1), 6), 25); /* Michael Somos, Apr 01 2015 */

E := proc(k) local n,t1; t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n,n=1..60); series(t1,q,60); end; E(6);
# alternative
A013973 := proc(n)
    if n = 0 then
        1;
    else
        -504*numtheory[sigma][5](n) ;
    end if;
end proc:
seq(A013973(n),n=0..10) ; # R. J. Mathar, Feb 22 2021

a[ n_] := If[ n < 1, Boole[n == 0], -504 DivisorSigma[ 5, n]]; (* Michael Somos, Apr 21 2013 *)
a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, t2^3 - 33 (t2 + t3) t2 t3 + t3^3], {q, 0, n}]; (* Michael Somos, Apr 21 2013 *)
a[ n_] := SeriesCoefficient[ With[ {t3 = EllipticTheta[ 3, 0, q]^4, t4 = EllipticTheta[ 4, 0, q]^4}, (t3^3 - 3 (t3 - t4)^2 (t3 + t4) + t4^3) / 2], {q, 0, 2 n}]; (* Michael Somos, Jun 04 2014 *)
a[ n_] := SeriesCoefficient[ With[ {e1 = QPochhammer[ q]^8, e4 = 32 q QPochhammer[ q^4]^8}, (e1 + e4) (e1^2 - 16 e1 e4 - 8 e4^2) / QPochhammer[ q^2]^12], {q, 0, n}]; (* Michael Somos, Apr 01 2015 *)
a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, t2^3 - 3/2 (t2 + t3) t2 t3 + t3^3], {q, 0, 2 n}]; (* Michael Somos, Jul 31 2016 *)
terms = 25; E6[x_] = 1-(12/BernoulliB[6])*Sum[k^5*x^k/(1-x^k), {k, terms}]; CoefficientList[E6[x] + O[x]^terms, x] (* Jean-François Alcover, Feb 28 2018 *)

{a(n) = if( n<1, n==0, -504 * sigma( n, 5))};

{a(n) = my(A, A1, A4); if( n<0, 0, A = x * O(x^n); A1 = eta(x + A)^8; A4 = 32 * x * eta(x^4 + A)^8; polcoeff( (A1 + A4) * (A1^2 - 16 * A1 * A4 - 8 * A4^2) / eta(x^2 + A)^12, n))}; /* Michael Somos, Dec 30 2008 */

ModularForms( Gamma1(1), 6, prec=25).0; # Michael Somos, Jun 04 2013

E6(q) = 1 - 504*Sum_{i>=1} sigma_5(i)q^i where sigma_5(n) is A001160, the sum of fifth powers of the divisors of n. It can also be expressed as E6(q) = 1 - 504*Sum_{i>=1} i^5*q^i/(1-q^i). - Gene Ward Smith, Aug 22 2006
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2*v - 8*u^2*w - 66*u*v^2 + 592*u*v*w - 512*u*w^2 + 121*v^3 - 4224*v^2*w + 4096*v*w^2. - Michael Somos, Apr 10 2005
Expansion of Ramanujan's function R(q) = 216*g3 (Weierstrass invariant).
Expansion of (eta(q)^8 + 32 * eta(q^4)^8) * (eta(q)^16 - 512 * eta(q)^8 * eta(q^4)^8 - 8192 * eta(q^4)^16) / eta(q^2)^12 in powers of q. - Michael Somos, Dec 30 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / t) = - (t/i)^6 * f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 30 2008
E6(q) = eta(q)^24 / eta(q^2)^12 - 480 * eta(q^2)^12 - 16896 * eta(q^2)^12 * eta(q^4)^8 / eta(q)^8 + 8192 * eta(q^4)^24 / eta(q^2)^12. - Seiichi Manyama, May 08 2017

A282050 Coefficients in q-expansion of (E_4^2 - E_2*E_6)/1008, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 66, 732, 4228, 15630, 48312, 117656, 270600, 533637, 1031580, 1771572, 3094896, 4826822, 7765296, 11441160, 17318416, 24137586, 35220042, 47045900, 66083640, 86124192, 116923752, 148035912, 198079200, 244218775, 318570252, 389021400, 497449568, 594823350

Offset: 0

Seiichi Manyama, Feb 05 2017

Multiplicative because A001160 is. - Andrew Howroyd, Jul 23 2018

			a(6) = 1^6*6^1 + 2^6*3^1 + 3^6*2^1 + 6^6*1^1 = 48312.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), this sequence (phi_{6, 1}), A282060 (phi_{8, 1}).
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A145095 (-q*E'_6), A008410 (E_4^2 = E_8), A282096 (E_2*E_6).
Cf. A001160, A013664, A131472.

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}];
(E4[x]^2 - E2[x]*E6[x])/1008 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)
Table[n * DivisorSigma[5, n], {n, 0, 30}] (* Amiram Eldar, Sep 06 2023 *)
nmax = 40; CoefficientList[Series[x*Sum[k^6*x^(k-1)/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 01 2025 *)

a(n) = if(n < 1, 0, n * sigma(n, 5)); \\ Andrew Howroyd, Jul 23 2018

a(n) = A145095(n)/504 for n > 0.
G.f.: phi_{6, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
a(n) = (A008410(n) - A282096(n))/1008. - Seiichi Manyama, Feb 10 2017
If p is a prime, a(p) = p^6 + p = A131472(p). - Seiichi Manyama, Feb 10 2017
a(n) = n*A001160(n) for n > 0. - Seiichi Manyama, Feb 18 2017
Sum_{k=1..n} a(k) ~ zeta(6) * n^7 / 7. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(5*e+5)-1)/(p^5-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-6). (End)
G.f. Sum_{k>=1} k^6*x^(k-1)/(1 - x^k)^2. - Vaclav Kotesovec, Aug 01 2025

A145094 Coefficients in expansion of Eisenstein series q*E'_4.

Original entry on oeis.org

240, 4320, 20160, 70080, 151200, 362880, 577920, 1123200, 1635120, 2721600, 3516480, 5886720, 6857760, 10402560, 12700800, 17975040, 20049120, 29432160, 31281600, 44150400, 48545280, 63296640, 67167360, 94348800, 94506000, 123439680, 132451200

Offset: 1

N. J. A. Sloane, Feb 28 2009

nonn

			G.f. = 240*q + 4320*q^2 + 20160*q^3 + 70080*q^4 + 151200*q^5 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..1000
M. Kaneko and D. Zagier, Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998.
Eric Weisstein's World of Mathematics, Eisenstein Series

Cf. A004009, A076835.

Cf. A076835 (-q*E'_2), this sequence (q*E'_4), A145095 (-q*E'_6).

terms = 28;
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])/3 + O[x]^terms // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Feb 23 2018 *)
nmax = 40; Rest[CoefficientList[Series[240*x*Sum[k^4*x^(k-1)/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 01 2025 *)

q*E'_4 = (E_2*E_4-E_6)/3.
G.f.: 240*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) ~  8 * Pi^4 * n^5 / 15. - Vaclav Kotesovec, May 09 2022

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

N. J. A. Sloane, Feb 28 2009

nonn

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

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Masanobu Kaneko and Don Zagier, Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998.
Eric Weisstein's World of Mathematics, Eisenstein Series.

Cf. A006352 (E_2), A004009 (E_4), A064987.

Cf. this sequence (-q*E'_2), A145094 (q*E'_4), A145095 (-q*E'_6).

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);

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 *)

a(n) = 24 * n * sigma(n); \\ Amiram Eldar, Jan 07 2025

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

A289637 Coefficients in expansion of -q*E'_6/E_6 where E_6 is the Eisenstein Series (A013973).

Original entry on oeis.org

504, 287280, 153540576, 82226602080, 44031499226064, 23578504122108096, 12626092121367162816, 6761166974864088760512, 3620548496603402008959384, 1938773508354916749345180960, 1038197035676506069321210300320

Offset: 1

Seiichi Manyama, Jul 09 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..366

-q*E'_k/E_k: A289635 (k=2), A289636 (k=4), this sequence (k=6), A289638 (k=8), A289639 (k=10), A289640 (k=14).

Cf. A000706, A006352 (E_2), A013973 (E_6), A145095, A288851.

nmax = 20; Rest[CoefficientList[Series[504*x*Sum[k*DivisorSigma[5, k]*x^(k-1), {k, 1, nmax}]/(1 - 504*Sum[DivisorSigma[5, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 09 2017 *)

a(n) = Sum_{d|n} d * A288851(d).
a(n) = A288840(n)/2 + 12*A000203(n).
a(n) = -Sum_{k=1..n-1} A013973(k)*a(n-k) - A013973(n)*n.
G.f.: 1/2 * E_8/E_6 - 1/2 * E_2.
a(n) ~ exp(2*Pi*n). - Vaclav Kotesovec, Jul 09 2017

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A013973 Expansion of Eisenstein series E_6(q) (alternate convention E_3(q)).

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A282050 Coefficients in q-expansion of (E_4^2 - E_2*E_6)/1008, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A145094 Coefficients in expansion of Eisenstein series q*E'_4.

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

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A289637 Coefficients in expansion of -q*E'_6/E_6 where E_6 is the Eisenstein Series (A013973).

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

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

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