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A008410 a(0) = 1, a(n) = 480*sigma_7(n). 44
1, 480, 61920, 1050240, 7926240, 37500480, 135480960, 395301120, 1014559200, 2296875360, 4837561920, 9353842560, 17342613120, 30119288640, 50993844480, 82051050240, 129863578080, 196962563520 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

REFERENCES

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.

H. D. Nguyen, D. Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013; http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.391.2522&rep=rep1&type=pdf. Mentions this sequence. - From N. J. A. Sloane, Mar 16 2014

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.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Eisenstein Series.

Index entries for sequences related to Eisenstein series

FORMULA

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

EXAMPLE

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

MAPLE

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);

MATHEMATICA

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 *)

PROG

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

(PARI) {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 */

(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)^2, n))}; /* Michael Somos, Dec 30 2008 */

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

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

CROSSREFS

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.

Sequence in context: A022047 A107511 A288471 * A305569 A289638 A289744

Adjacent sequences:  A008407 A008408 A008409 * A008411 A008412 A008413

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified April 7 19:24 EDT 2020. Contains 333306 sequences. (Running on oeis4.)