A029829 -id:A029829 - cp's OEIS frontend

A004009 Expansion of Eisenstein series E_4(q) (alternate convention E_2(q)); theta series of E_8 lattice.

1, 240, 2160, 6720, 17520, 30240, 60480, 82560, 140400, 181680, 272160, 319680, 490560, 527520, 743040, 846720, 1123440, 1179360, 1635120, 1646400, 2207520, 2311680, 2877120, 2920320, 3931200, 3780240, 4747680, 4905600, 6026880

Offset: 0

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
E_8 is also the Barnes-Wall lattice in 8 dimensions.
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan Lambert series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).
The E_8 lattice is integral, unimodular, and even. The 240 shortest nonzero vectors in the lattice have norm squared 2. Of these vectors, 128 are all half-integer, and 112 are all integer. - Michael Somos, Jun 10 2019

			G.f. = 1 + 240*x + 2160*x^2 + 6720*x^3 + 17520*x^4 + 30240*x^5 + 60480*x^6 + ...
G.f. = 1 + 240*q^2 + 2160*q^4 + 6720*q^6 + 17520*q^8 + 30240*q^10 + 60480*q^12 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 123.
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.
N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 111.
S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 (1916), 11-15 (Eq. (25)). Collected Papers of Srinivasa Ramanujan, Chap. 16, Ed. G. H. Hardy et al., Chelsea, NY, 1962.
S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 (1916), 11-15 (Eq. (25)). Ramanujan's Papers, p. 196, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.
Jean-Pierre Serre, "A Course in Arithmetic", Springer, 1978
Joseph H. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves", Springer, 1994
N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from N. J. A. Sloane)
D. Bump, Automorphic Forms and Representations, Cambr. Univ. Press, 1997, p. 29.
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
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 Q(q).
Henry Cohn and Stephen D. Miller, Some properties of optimal functions for sphere packing in dimensions 8 and 24, arXiv:1603.04759 [math.MG], 2016.
H. S. M. Coxeter, Integral Cayley numbers, Duke Math. J. 13 (1946), 561-578; reprinted in "Twelve Geometric Essays", pp. 20-39.
D. de Laat and F. Vallentin, A Breakthrough in Sphere Packing: The Search for Magic Functions, arXiv preprint arXiv:1607.02111 [math.MG], 2016.
Yang-Hui He and John McKay, Sporadic and Exceptional, arXiv:1505.06742 [math.AG], 2015.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
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
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.]
Robert V. Moody and Jiri Patera, Fast recursion formula for weight multiplicities, Bulletin of the American Mathematical Society 7.1 (1982): 237-242.
G. Nebe and N. J. A. Sloane, Home page for E_8 lattice
H. Ochiai, Counting functions for branched covers of elliptic curves and quasi-modular forms, arXiv:math-ph/9909023, 1999.
S. Ramanujan, On the coefficients in the expansions of certain modular functions, Proc. Royal Soc., A, 95 (1918), 144-155.
Michael Somos, Introduction to Ramanujan theta functions
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Seven Staggering Sequences.
Maryna S. Viazovska, The sphere packing problem in dimension 8, arXiv preprint arXiv:1603.04246 [math.NT], 2016.
Martin H. Weissman, Octonions, Cubes, Embeddings, March 2, 2009.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Eisenstein Series.
Eric Weisstein's World of Mathematics, Leech Lattice.
Eric Weisstein's World of Mathematics, Barnes-Wall Lattice
Wikipedia, Eisenstein series
Wikipedia, E_8 lattice
Index entries for sequences related to Eisenstein series
Index entries for sequences related to Barnes-Wall lattices

Cf. A046948 (partial sums), A000143, A108091 (eighth root).
Cf. A008410, A008411, A001158.
Cf. A006352 (E_2), 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. A007331 (theta_2(q)^8 / 256), A000143 (theta_3(q)^8), A035016 (theta_4(q)^8).

Basis( ModularForms( Gamma1(1), 4), 29) [1]; /* Michael Somos, May 11 2015 */

L := Lattice("E",8); A := ThetaSeries(L, 57); A; /* Michael Somos, Jun 10 2019 */

with(numtheory); 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(4);

a[ n_] := If[ n < 1, Boole[n == 0], 240 DivisorSigma[ 3, n]]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, t2^2 + 14 t2 t3 + t3^2], {q, 0, n}]; (* Michael Somos, Jun 04 2014 *)
max = 30; s = 1 + 240*Sum[k^3*(q^k/(1 - q^k)), {k, 1, max}] + O[q]^max; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, after Gene Ward Smith *)
a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, t2^2 - t2 t3 + t3^2], {q, 0, 2 n}]; (* Michael Somos, Jul 31 2016 *)

{a(n) = if( n<1, n==0, 240 * sigma(n, 3))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^24 + 256 * x * eta(x^2 + A)^24) / (eta(x + A) * eta(x^2 + A))^8, n))}; /* Michael Somos, Dec 30 2008 */

q='q+O('q^50); Vec((eta(q)^24+256*q*eta(q^2)^24)/(eta(q)*eta(q^2))^8) \\ Altug Alkan, Sep 30 2018

from sympy import divisor_sigma
def a(n): return 1 if n == 0 else 240 * divisor_sigma(n, 3)
[a(n) for n in range(51)]  # Indranil Ghosh, Jul 15 2017

ModularForms(Gamma1(1), 4, prec=30).0 ; # Michael Somos, Jun 04 2013

Can also be expressed as E4(q) = 1 + 240*Sum_{i >= 1} i^3 q^i/(1 - q^i) - Gene Ward Smith, Aug 22 2006
Theta series of E_8 lattice = 1 + 240 * Sum_{m >= 1} sigma_3(m) * q^(2*m), where sigma_3(m) is the sum of the cubes of the divisors of m (A001158).
Expansion of (phi(-q)^8 - (2 * phi(-q) * phi(q))^4 + 16 * phi(q)^8) in powers of q where phi() is a Ramanujan theta function.
Expansion of (eta(q)^24 + 256 * eta(q^2)^24) / (eta(q) * eta(q^2))^8 in powers of q. - Michael Somos, Dec 30 2008
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + 33*v^2 + 256*w^2 - 18*u*v + 16*u*w - 288*v*w . - Michael Somos, Jan 05 2006
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 + 16*u2^2 + 81*u3^2 + 1296*u6^2 - 14*u1*u2 - 18*u1*u3 + 30*u1*u6 + 30*u2*u3 - 288*u2*u6 - 1134*u3*u6 . - Michael Somos, Apr 15 2007
G.f. A(x) satisfies 0 = f(A(x), A(x^3), A(x^9)) where f(u, v, w) = u^3*v + 9*w*u^3 - 84*u^2*v^2 + 246*u*v^3 - 253*v^4 - 675*w*u^2*v + 729*w^2*u^2 - 4590*w*u*v^2 + 19926*w*v^3 - 54675*w^2*u*v + 59049*w^3*u + 531441*w^3*v - 551124*w^2*v^2 . - Michael Somos, Apr 15 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / t) = (t/i)^4 * f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 30 2008
Convolution square is A008410. A008411 is convolution of this sequence with A008410.
Expansion of Ramanujan's function Q(q^2) = 12 (omega/Pi)^4 g2 (Weierstrass invariant) in powers of q^2.
Expansion of a(q) * (a(q)^3 + 8*c(q)^3) in powers of q where a(), c() are cubic AGM theta functions. - Michael Somos, Jan 14 2015
G.f. is (theta_2(q)^8 + theta_3(q)^8 + theta_4(q)^8) / 2 where q = exp(Pi i t). So a(n) = A008430(n) + 128*A007331(n) (= A000143(2*n) + 128*A007331(n) = A035016(2*n) + 128*A007331(n)). - Seiichi Manyama, Sep 30 2018
a(n) = 240*A001158(n) if n>0. - Michael Somos, Oct 01 2018
Sum_{k=1..n} a(k) ~ 2 * Pi^4 * n^4 / 3. - Vaclav Kotesovec, Jan 14 2024

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

Original entry on oeis.org

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

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

Original entry on oeis.org

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

A008410 a(0) = 1, a(n) = 480*sigma_7(n).

Original entry on oeis.org

1, 480, 61920, 1050240, 7926240, 37500480, 135480960, 395301120, 1014559200, 2296875360, 4837561920, 9353842560, 17342613120, 30119288640, 50993844480, 82051050240, 129863578080, 196962563520

Offset: 0

N. J. A. Sloane

nonn

Eisenstein series E_8(q) (alternate convention E_4(q)); theta series of direct sum of 2 copies of E_8 lattice.

			G.f. = 1 + 480*q + 61920*q^2 + 1050240*q^3 + 7926240*q^4 + 37500480*q^5 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 123.
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 p. 111.
S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 (1916), 11-15 (Eq. (25)). Collected Papers of Srinivasa Ramanujan, Chap. 16, Ed. G. H. Hardy et al., Chelsea, NY, 1962.
S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 (1916), 11-15 (Eq. (25)). Ramanujan's Papers, p. 196, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
H. D. Nguyen, D. Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013; Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014
Eric Weisstein's World of Mathematics, Eisenstein Series.
Index entries for sequences related to Eisenstein series

Cf. A013973.
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).
Convolution square of A004009.

Basis( ModularForms( Gamma1(1), 8), 33) [1]; /* Michael Somos, May 27 2014 */

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(8);

a[ n_] := If[ n < 1, Boole[n == 0], 480 DivisorSigma[ 7, n]]; (* Michael Somos, Jun 04 2013 *)
nmax = 60; CoefficientList[Series[(Product[(1-x^k)^8 / (1+x^k)^8, {k, 1, nmax}] + 256 * x * Product[(1+x^k)^16 *(1-x^k)^8, {k, 1, nmax}])^2, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 02 2017 *)

{a(n) = if( n<1, n==0, 480 * sigma(n, 7))};

{a(n) = local(A, e1, e2, e4); if( n<0, 0, n*=2; A = x * O(x^n); e1 = eta(x + A)^16; e2 = eta(x^2 + A)^16; e4 = eta(x^4 + A)^16; polcoeff( (e1*e2^3 + 256*x^2 * e4*(e2^3 + e1^2*e4)) / (e1*e2*e4), n))}; /* Michael Somos, Jun 29 2005 */

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x + A)^24 + 256 * x * eta(x^2 + A)^24) / (eta(x + A) * eta(x^2 + A))^8)^2, n))}; /* Michael Somos, Dec 30 2008 */

ModularForms( Gamma1(1), 8, prec=33).0; # Michael Somos, Jun 04 2013

Equivalently, g.f. = (theta2^16+theta3^16+theta4^16)/2.
G.f. Sum{k>=0} a(k)q^(2k) = (theta2^16+theta3^16+theta4^16)/2.
Expansion of ((eta(q)^24 + 256 * eta(q^2)^24) / (eta(q) * eta(q^2))^8)^2 in powers of q. - Michael Somos, Dec 30 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / t) = (t/i)^8 * f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 30 2008
a(n) = 480*A013955(n). - R. J. Mathar, Oct 10 2012

A013974 Eisenstein series E_10(q) (alternate convention E_5(q)).

Original entry on oeis.org

1, -264, -135432, -5196576, -69341448, -515625264, -2665843488, -10653352512, -35502821640, -102284205672, -264515760432, -622498190688, -1364917062432, -2799587834736, -5465169838656, -10149567696576, -18177444679944

Offset: 0

N. J. A. Sloane

sign

			G.f. = 1 - 264*q - 135432*q^2 - 5196576*q^3 - 69341448*q^4 - 515625264*q^5 + ...

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 p. 111.

T. D. Noe, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Eisenstein Series.
Index entries for sequences related to Eisenstein series

Cf. A008410.
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).
Convolution of A004009 and A013973.

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(10);

a[ n_] := If[ n < 1, Boole[n == 0], -264 DivisorSigma[ 9, n]]; (* Michael Somos, Jun 04 2013 *)
a[ n_] := SeriesCoefficient[ With[{t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, t2^5 - 19 t2 t3 (t2^3 + t3^3) - 494 (t2 t3)^2 (t2 + t3) + t3^5], {q, 0, n}]; (* Michael Somos, Jun 04 2013 *)
terms = 17; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[Ei[10] + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)

{a(n) = if( n<1, n==0, -264 * sigma( n, 9))};

ModularForms( Gamma1(1), 10, prec=13).0; # Michael Somos, Jun 04 2013

Sum_{n >= 0} a(n)/exp(Pi)^(2n) = 0 or is very close to 0. - Gerald McGarvey, Jan 25 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / t) = - (t/i)^10 * f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 30 2008
G.f.: 1 - 264*Sum_{k>=1} k^9*x^k/(1 - x^k). - Ilya Gutkovskiy, Aug 31 2017

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A004009 Expansion of Eisenstein series E_4(q) (alternate convention E_2(q)); theta series of E_8 lattice.

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

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

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A008410 a(0) = 1, a(n) = 480*sigma_7(n).

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A013974 Eisenstein series E_10(q) (alternate convention E_5(q)).

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