login
This site is supported by donations to The OEIS Foundation.

 

Logo

110 people attended OEIS-50 (videos, suggestions); annual fundraising drive to start soon (donate); editors, please edit! (stack is over 300), your editing is more valuable than any donation.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A004009 Expansion of Eisenstein series E_4(q) (alternate convention E_2(q)); theta series of E_8 lattice.
(Formerly M5416)
53
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

Expansion of Ramanujan's function Q(q) = 12 g2 (Weierstrass invariant).

REFERENCES

D. Bump, Automorphic Forms..., Camb., 1997 p. 29.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 123.

H. S. M. Coxeter, Integral Cayley numbers, Duke Math. J. 13 (1946), 561-578; reprinted in "Twelve Geometric Essays", pp. 20-39.

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.

Masao Koike, Modular forms on non-compact arithmetic triangle groups, preprint.

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).

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..1000

H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

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

S. Ramanujan, On the coefficients in the expansions of certain modular functions, Proc. Royal Soc., A, 95 (1918), 144-155.

M. 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.

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

Index entries for sequences related to Eisenstein series

Index entries for sequences related to Barnes-Wall lattices

FORMULA

Can also be expressed as E4(q) = 1 + 240 sum_{i=1}^infinity i^3 q^i/(1-q^i) - Gene Ward Smith, Aug 22 2006

Theta series of E_8 lattice = 1 + 240 * Sum ( sigma_3 (m) * q^2m ), m = 1..inf, 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.

EXAMPLE

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

MAPLE

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);

MATHEMATICA

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 *)

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q]^4 + EllipticTheta[ 3, 0, q]^4 + EllipticTheta[ 4, 0, q]^4) / 2], {q, 0, 2 n}; (* Michael Somos, Jun 04 2014 *)

PROG

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

(PARI) {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, n))}; /* Michael Somos, Dec 30 2008 */

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

CROSSREFS

Cf. A046948, A000143, A108091 (eighth root).

Cf. A008410, A008411, A001158.

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).

Sequence in context: A205264 A205257 A049335 * A180277 A005950 A004536

Adjacent sequences:  A004006 A004007 A004008 * A004010 A004011 A004012

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified October 31 06:33 EDT 2014. Contains 248845 sequences.