A076835 -id:A076835 - cp's OEIS frontend

A006352 Coefficients in expansion of Eisenstein series E_2 (also called E_1 or G_2).

1, -24, -72, -96, -168, -144, -288, -192, -360, -312, -432, -288, -672, -336, -576, -576, -744, -432, -936, -480, -1008, -768, -864, -576, -1440, -744, -1008, -960, -1344, -720, -1728, -768, -1512, -1152, -1296, -1152, -2184, -912, -1440, -1344, -2160, -1008, -2304, -1056, -2016, -1872, -1728

Offset: 0

N. J. A. Sloane

Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

The series Q(q), R(q) are modular forms, but P(q) is not. - Michael Somos, May 18 2017

			G.f. = 1 - 24*x - 72*x^2 - 96*x^3 - 168*x^4 - 144*x^5 - 288*x^6 + ...

R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 53.
N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see pp. 111 and 113.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Zagier, Don. "Elliptic modular forms and their applications." The 1-2-3 of modular forms. Springer Berlin Heidelberg, 2008. 1-103. See p. 19, Eq. (17).

T. D. Noe, Table of n, a(n) for n = 0..1000
F. Beukers, Another congruence for the Apéry numbers, J. Number Theory 25 (1987), no. 2, 201-210.
J. H. Bruinier and K. Ono, Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms.
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 P(q).
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.
H. Ochiai, Counting functions for branched covers of elliptic curves and quasi-modular forms, arXiv:math-ph/9909023, 1999.
V. P. Varin, Special solutions to Chazy equation, Comput. Math. Math. Phys. 57, No. 2, 211-235 (2017), eq (75).
Eric Weisstein's World of Mathematics, Eisenstein Series.
Index entries for sequences related to Eisenstein series.

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. A000594 (Delta), A076835, A145155 (Delta').

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(2);

a[n_] := -24*DivisorSigma[1, n]; a[0] = 1; Table[a[n], {n, 0, 46}] (* Jean-François Alcover, Dec 12 2012 *)
a[ n_] := If[ n < 1, Boole[n == 0], -24 DivisorSigma[ 1, n]]; (* Michael Somos, Apr 08 2015 *)

{a(n) = if( n<1, n==0, -24 * sigma(n))}; /* Michael Somos, Apr 09 2003 */

from sympy import divisor_sigma
def a(n): return 1 if n == 0 else -24 * divisor_sigma(n)
[a(n) for n in range(51)]  # Indranil Ghosh, Jul 15 2017

a(n) = -24*sigma(n) = -24*A000203(n), for n>0.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2 + 4*u2^2 + 9*u3^2 + 36*u6^2 - 8*u1*u2 + 6*u1*u3 + 24*u2*u6 - 72*u3*u6. - Michael Somos, May 29 2005
G.f.: 1 - 24*sum(k>=1, k*x^k/(1 - x^k)).
G.f.: 1 + 24 *x*deriv(eta(x))/eta(x) where eta(x) = prod(n>=1, 1-x^n); (cf. A000203). - Joerg Arndt, Sep 28 2012
G.f.: 1 - 24*x/(1-x) + 48*x^2/(Q(0) -  2*x^2 + 2*x), where Q(k)= (2*x^(k+2) - x - 1)*k - 1 - 2*x + 3*x^(k+2) - x*(k+1)*(k+3)*(1-x^(k+2))^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 16 2013
G.f.: q*Delta'/Delta where Delta is the generating function of Ramanujan's tau function (A000594). - Seiichi Manyama, Jul 15 2017

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

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

N. J. A. Sloane, Feb 28 2009

nonn

			G.f. = 504*q + 33264*q^2 + 368928*q^3 + 2130912*q^4 + 7877520*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. A076835 (-q*E'_2), A145094 (q*E'_4), this sequence (-q*E'_6).

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 *)

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

A289635 Coefficients in expansion of -q*E'_2/E_2 where E_2 is the Eisenstein Series (A006352).

Original entry on oeis.org

24, 720, 19296, 517920, 13893264, 372707136, 9998360256, 268219317312, 7195339794744, 193024557070560, 5178140391612960, 138910500937231488, 3726458885094926160, 99967214347459657344, 2681753442755678231616

Offset: 1

Seiichi Manyama, Jul 09 2017

nonn

			a(1) = - A006352(1)*1 = 24,
a(2) = -(A006352(1)*a(1)) - A006352(2)*2 = 720,
a(3) = -(A006352(1)*a(2)  + A006352(2)*a(1)) - A006352(3)*3 = 19296,
a(4) = -(A006352(1)*a(3)  + A006352(2)*a(2)  + A006352(3)*a(1)) - A006352(4)*4 = 517920.

Seiichi Manyama, Table of n, a(n) for n = 1..700

-q*E'_k/E_k: this sequence (k=2), A289636 (k=4), A289637 (k=6), A289638 (k=8), A289639 (k=10), A289640 (k=14).

Cf. A000203, A006352 (E_2), A076835, A211342, A288816, A288877, A288968.

nmax = 20; Rest[CoefficientList[Series[24*x*Sum[k*DivisorSigma[1, k]*x^(k-1), {k, 1, nmax}]/(1 - 24*Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 09 2017 *)

a(n) = Sum_{d|n} d * A288968(d).
a(n) = A288877(n)/12 + 2*A000203(n).
a(n) = -Sum_{k=1..n-1} A006352(k)*a(n-k) - A006352(n)*n.
G.f.: 1/12 * E_4/E_2 - 1/12 * E_2.
a(n) ~ 1 / r^n, where r = A211342 = 0.037276810296451658150980785651644618... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24. - Vaclav Kotesovec, Jul 09 2017

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

Seiichi Manyama, Jul 11 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

(-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.

Table[480*n*DivisorSigma[7, n], {n, 1, 20}] (* Jean-François Alcover, Feb 27 2018 *)

a(n) = 480 * n * sigma(n, 7); \\ Amiram Eldar, Jan 07 2025

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

Seiichi Manyama, Jul 11 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

(-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.

Table[264*n*DivisorSigma[9, n], {n, 1, 18}] (* Jean-François Alcover, Feb 27 2018 *)

a(n) = 264*n*sigma(n, 9); \\ Michel Marcus, Feb 27 2018

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

Seiichi Manyama, Jul 11 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

(-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.

Table[24*n*DivisorSigma[13, n], {n, 1, 15}] (* Jean-François Alcover, Feb 27 2018 *)

a(n) = 24*n*sigma(n, 13); \\ Michel Marcus, Feb 27 2018

a(n) = 24*A282597(n) = 24*n*A013961(n).

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

Seiichi Manyama, Feb 07 2017

nonn

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

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.

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 *)

-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).

cp's OEIS Frontend

A006352 Coefficients in expansion of Eisenstein series E_2 (also called E_1 or G_2).

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

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

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A289635 Coefficients in expansion of -q*E'_2/E_2 where E_2 is the Eisenstein Series (A006352).

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

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

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