A108091 -id:A108091 - 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

A106205 Expansion of (q*j(q))^(1/24) where j(q) is the elliptic modular invariant (A000521).

Original entry on oeis.org

1, 31, -2848, 413823, -68767135, 12310047967, -2309368876639, 447436508910495, -88755684988520798, 17924937024841839390, -3671642907594608226078, 760722183234128461061246, -159105706560247952472114973

Offset: 0

Michael Somos, Apr 25 2005

sign

From Vaclav Kotesovec, Jun 10 2018: (Start)
For k > 0, if mod(k,8) <> 0 then (q*j(q))^(k/24) is asymptotic to -(-1)^n * sin(k*Pi/8) * k * 3^(k/8) * Gamma(1/3)^(3*k/4) * Gamma(k/8) * exp(Pi*sqrt(3)*n) / (Pi^(k/2 + 1) * 2^(k/8 + 3) * exp(k*Pi/(8*sqrt(3))) * n^(k/8 + 1)). Equivalently, is asymptotic to -(-1)^n * k * 3^(k/8) * Gamma(1/3)^(3*k/4) * exp(Pi*sqrt(3)*(n - k/24)) / (Pi^(k/2) * 2^(k/8 + 3) * Gamma(1 - k/8) * n^(k/8 + 1)).
For k > 0, if mod(k,8) = 0 then (q*j(q))^(k/24) is asymptotic to exp(Pi*sqrt(2*k*n/3)) * k^(1/4) / (2^(5/4) * 3^(1/4) * n^(3/4)).
(End)

			1 + 31*q - 2848*q^2 + 413823*q^3 - 68767135*q^4 + 12310047967*q^5 - 2309368876639*q^6 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..424

(q*j(q))^(k/24): this sequence (k=1), A289297 (k=2), A289298 (k=3), A289299 (k=4), A289300 (k=5), A289301 (k=6), A289302 (k=7), A007245 (k=8), A289303 (k=9), A289304 (k=10), A289305 (k=11), A161361 (k=12).

Cf. A000521, A192731.

CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^(1/8) / (2*QPochhammer[-1, x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 23 2017 *)
(q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(1/24) + O[q]^13 // CoefficientList[#, q]& (* Jean-François Alcover, Nov 02 2017 *)

{a(n)=if(n<0,0, polcoeff( (ellj(x+x^2*O(x^n))*x)^(1/24),n))}

This is essentially the eighth root of the theta series of E_8 (A108091), divided by the Dedekind eta function. - N. J. A. Sloane, Aug 08 2005
G.f.: Product_{n>=1} (1-q^n)^(A192731(n)/24). - Seiichi Manyama, Jul 02 2017
a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(9/8), where c = 0.11364889078525240958152388212499254894082832445224690827436413842337... = 3^(1/8) * sqrt(2 - sqrt(2)) * Gamma(1/8) * Gamma(1/3)^(3/4) / (2^(33/8) * exp(Pi/(8 * sqrt(3))) * Pi^(3/2)). - Vaclav Kotesovec, Jul 02 2017, updated Mar 06 2018
a(n) * A289397(n) ~ c * exp(2*Pi*sqrt(3)*n) / n^2, where c = -sqrt(2-sqrt(2)) / (16*Pi). - Vaclav Kotesovec, Mar 06 2018

A289368 Coefficients in expansion of (E_6^2/E_4^3)^(1/24).

Original entry on oeis.org

1, -72, -6048, -4217184, -1264437504, -606533479920, -251777443450752, -117085712395216320, -53634689421870422016, -25408429618361083967592, -12110787335129301116994240, -5854620911089647830793873696

Offset: 0

Seiichi Manyama, Jul 04 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..367

(E_6^2/E_4^3)^(k/288): A289366 (k=1), A296609 (k=2), A296614 (k=3), A296652 (k=4), A297021 (k=6), A299422 (k=8), A299862 (k=9), this sequence (k=12), A299856 (k=16), A299857 (k=18), A299858 (k=24), A299863 (k=32), A299859 (k=36), A299860 (k=48), A299861 (k=72), A299414 (k=96), A299413 (k=144), A289210 (k=288).
Cf. A000521 (j), A108091 (E_4^(1/8)), A109817 (E_6^(1/12)).
Cf. A289366, A289367, A294974, A294976.

nmax = 20; CoefficientList[Series[((1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^2 / (1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^3)^(1/24), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

G.f.: (1 - 1728/j)^(1/24).
G.f.: Product_{n>=1} (1-q^n)^(12*A289367(n)).
a(n) ~ c * exp(2*Pi*n) / n^(13/12), where c = -Gamma(1/4)^(1/3) / (2^(7/3) * 3^(23/24) * Pi^(1/4) * Gamma(11/12)) = -0.07569217204117312767729284017524325060022536591050774997610261275428... - Vaclav Kotesovec, Jul 08 2017, updated Mar 04 2018
a(n) * A289369(n) ~ -(sqrt(3)-1) * exp(4*Pi*n) / (24*sqrt(2)*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A289369 Coefficients in expansion of (E_4^3/E_6^2)^(1/24).

Original entry on oeis.org

1, 72, 11232, 5461344, 2029222656, 924074630640, 411487620614784, 192705317913673152, 91031590937141544960, 43814578627107100088424, 21291642032558036150652480, 10450287314646252538819378464, 5166676457072455262194208351232

Offset: 0

Seiichi Manyama, Jul 04 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..367

(E_4^3/E_6^2)^(k/288): A289365 (k=1), A299694 (k=2), A299696 (k=3), A299697 (k=4), A299698 (k=6), A299943 (k=8), A299949 (k=9), this sequence (k=12), A299950 (k=16), A299951 (k=18), A299953 (k=24), A299993 (k=32), A299994 (k=36), A300052 (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).
Cf. A108091 (E_4^(1/8)), A109817 (E_6^(1/12)).
Cf. A289367, A294978, A294979.

nmax = 20; CoefficientList[Series[((1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^3 / (1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^2)^(1/24), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(-12*A289367(n)).
a(n) ~ c * exp(2*Pi*n) / n^(11/12), where c = 2^(1/3) * Pi^(1/4) / (3^(1/24) * Gamma(1/12) * Gamma(1/4)^(1/3)) =  0.0907014320494145997187363667820553893... - Vaclav Kotesovec, Jul 08 2017, updated Mar 04 2018
a(n) * A289368(n) ~ -(sqrt(3)-1) * exp(4*Pi*n) / (24*sqrt(2)*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A289292 Coefficients in expansion of E_4^(1/2).

Original entry on oeis.org

1, 120, -6120, 737760, -107249640, 17385063120, -3014720249760, 547287510713280, -102701836021530600, 19762301660609250840, -3878226140959368843120, 773209219953012480001440, -156173318001506652330786720, 31888935085481430265623676560

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..425
R. S. Maier, Nonlinear differential equations satisfied by certain classical modular forms, arXiv:0807.1081 [math.NT], 2008-2010, p. 34 equation (7.29b).

E_k^(1/2): A289291 (k=2), this sequence (k=4), A289293 (k=6), A004009 (k=8), A289294 (k=10), A289295 (k=14).
E_4^(k/8): A108091 (k=1), A289307 (k=2), A289308 (k=3), this sequence (k=4), A289309 (k=5), A289318 (k=6), A289319 (k=7).
Cf. A001421, A004009 (E_4), A110163.

terms = 14;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E4[x]^(1/2) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

G.f.: Product_{n>=1} (1-q^n)^(A110163(n)/2).
a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(3/2), where c = 3*Gamma(1/3)^9 / (32*sqrt(2)*Pi^(13/2)) = 0.27646925986847687648926173728588572192308632719... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018
G.f.: 3F2(1/6, 1/2, 5/6; 1, 1; 1728/j) where j is the elliptic modular invariant (A000521). - Seiichi Manyama, Jul 07 2017

A289307 Coefficients in expansion of E_4^(1/4) in powers of q.

Original entry on oeis.org

1, 60, -4860, 660480, -105063420, 18206269560, -3328461434880, 631226199152640, -122944850563477500, 24436796345920143420, -4935178772322020730360, 1009598430837232126725120, -208736157503462405753487360, 43541664791244563211024015480

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

			From _Seiichi Manyama_, Jul 07 2017: (Start)
2F1(1/12, 5/12; 1; 1728/j)
= 1 + (1*5)/(1*1) * 12/j + (1*5*13*17)/(1*1*2*2) * (12/j)^2 + (1*5*13*17*25*29)/(1*1*2*2*3*3) * (12/j)^3 + ...
= 1 + 60/j + 39780/j^2 + 38454000/j^3 + ...
= 1 + 60*q - 44640*q^2 + 21399120*q^3 - ...
           + 39780*q^2 - 59192640*q^3 + ...
                       + 38454000*q^3 - ...
                                      + ...
= 1 + 60*q -  4860*q^2 +   660480*q^3 - ... (End)

Seiichi Manyama, Table of n, a(n) for n = 0..424
M. Kontsevich and D. Zagier, Periods, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22. Published in B. Engquist and W. Schmid, editors, Mathematics Unlimited - 2001 and Beyond, 2 vols., Springer-Verlag, 2001, pp. 771-808, section 2.3. Example 3.
R. S. Maier, Nonlinear differential equations satisfied by certain classical modular forms, arXiv:0807.1081 [math.NT], 2008-2010, p. 34 equation (7.29a).

E_4^(k/8): A108091 (k=1), this sequence (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5), A289318 (k=6), A289319 (k=7).

Cf. A000521 (j), A004009 (E_4), A066395 (1/j), A092870, A110163, A289210.

a[ n_] := SeriesCoefficient[ ComposeSeries[ Series[ Hypergeometric2F1[ 1/12, 5/12, 1, q], {q, 0, n}], q^2 / Series[q^2 KleinInvariantJ[ Log[q]/(2 Pi I)], {q, 0, n}]], {q, 0, n}]; (* Michael Somos, Jun 21 2018 *)

G.f.: Product_{n>=1} (1-q^n)^(A110163(n)/4).
G.f.: 2F1(1/12, 5/12; 1; 1728/j) where j is the elliptic modular invariant (A000521). - Seiichi Manyama, Jul 06 2017 [See also the Kontsevich and Zagier link, where t = 1728/j = 1 - Sum_{k>=0} A289210(k)*q^k, with q = q(z) = exp(2*Pi*I*z), Im(z) > 0. - Wolfdieter Lang, May 27 2018]
a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(5/4), where c = sqrt(3) * Gamma(1/3)^(9/2) * Gamma(1/4) / (16 * 2^(3/4) * Pi^4) = 0.201967785736579402060958871696381229013432952780653381728912717635... - Vaclav Kotesovec, Jul 07 2017, updated Mar 04 2018

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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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A106205 Expansion of (q*j(q))^(1/24) where j(q) is the elliptic modular invariant (A000521).

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A289368 Coefficients in expansion of (E_6^2/E_4^3)^(1/24).

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A289369 Coefficients in expansion of (E_4^3/E_6^2)^(1/24).

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A289292 Coefficients in expansion of E_4^(1/2).

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A289307 Coefficients in expansion of E_4^(1/4) in powers of q.

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A289392 Coefficients in expansion of E_2^(1/4).

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Formula

A289308 Coefficients in expansion of E_4^(3/8).