0,2

E_8 is also the Barnes-Wall lattice in 8 dimensions.

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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.

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, 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.

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, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Barnes-Wall Lattice

Index entries for sequences related to Eisenstein series

Index entries for sequences related to Barnes-Wall lattices

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

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).

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

Expansion of (phi(-q)^8 -(2phi(-q)phi(q))^4 +16phi(q)^8) in powers of q where phi() is a Ramanujan theta function.

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

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. 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

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

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);

(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 */

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

Cf. A001158.

A008410 is convolution of this sequence with itself. A008411 is convolution of this sequence with 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).

Sequence in context: A158562 A157766 A049335 this_sequence A005950 A004536 A145094

Adjacent sequences: A004006 A004007 A004008 this_sequence A004010 A004011 A004012

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).