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%I M5416 #164 Jan 17 2024 09:12:29
%S 1,240,2160,6720,17520,30240,60480,82560,140400,181680,272160,319680,
%T 490560,527520,743040,846720,1123440,1179360,1635120,1646400,2207520,
%U 2311680,2877120,2920320,3931200,3780240,4747680,4905600,6026880
%N Expansion of Eisenstein series E_4(q) (alternate convention E_2(q)); theta series of E_8 lattice.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C E_8 is also the Barnes-Wall lattice in 8 dimensions.
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C Ramanujan Lambert series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).
%C 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
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 123.
%D W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 53.
%D R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 53.
%D N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 111.
%D 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.
%D 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.
%D Jean-Pierre Serre, "A Course in Arithmetic", Springer, 1978
%D Joseph H. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves", Springer, 1994
%D 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.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A004009/b004009.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from N. J. A. Sloane)
%H D. Bump, <a href="https://doi.org/10.1017/CBO9780511609572">Automorphic Forms and Representations</a>, Cambr. Univ. Press, 1997, p. 29.
%H H. H. Chan and C. Krattenthaler, <a href="http://arXiv.org/abs/math.NT/0407061">Recent progress in the study of representations of integers as sums of squares</a>, arXiv:math/0407061 [math.NT], 2004.
%H Heng Huat Chan, Shaun Cooper, and Pee Choon Toh, <a href="http://unimodular.net/archive/RamEisenstein.pdf">Ramanujan's Eisenstein series and powers of Dedekind's eta-function</a>, Journal of the London Mathematical Society 75.1 (2007): 225-242. See Q(q).
%H Henry Cohn and Stephen D. Miller, <a href="http://arxiv.org/abs/1603.04759">Some properties of optimal functions for sphere packing in dimensions 8 and 24</a>, arXiv:1603.04759 [math.MG], 2016.
%H H. S. M. Coxeter, <a href="http://dx.doi.org/10.1215/S0012-7094-46-01347-6">Integral Cayley numbers</a>, Duke Math. J. 13 (1946), 561-578; reprinted in "Twelve Geometric Essays", pp. 20-39.
%H D. de Laat and F. Vallentin, <a href="http://arxiv.org/abs/1607.02111">A Breakthrough in Sphere Packing: The Search for Magic Functions</a>, arXiv preprint arXiv:1607.02111 [math.MG], 2016.
%H Yang-Hui He and John McKay, <a href="http://arxiv.org/abs/1505.06742">Sporadic and Exceptional</a>, arXiv:1505.06742 [math.AG], 2015.
%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0509316">On the Integrality of n-th Roots of Generating Functions</a>, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%H M. Kaneko and D. Zagier, <a href="http://www2.math.kyushu-u.ac.jp/~mkaneko/papers/atkin.pdf">Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials</a>, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998
%H Masao Koike, <a href="https://oeis.org/A004016/a004016.pdf">Modular forms on non-compact arithmetic triangle groups</a>, 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 Robert V. Moody and Jiri Patera, <a href="https://doi.org/10.1090/S0273-0979-1982-15021-2">Fast recursion formula for weight multiplicities</a>, Bulletin of the American Mathematical Society 7.1 (1982): 237-242.
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/E8.html">Home page for E_8 lattice</a>
%H H. Ochiai, <a href="http://arXiv.org/abs/math-ph/9909023">Counting functions for branched covers of elliptic curves and quasi-modular forms</a>, arXiv:math-ph/9909023, 1999.
%H S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper37/page1.htm">On the coefficients in the expansions of certain modular functions</a>, Proc. Royal Soc., A, 95 (1918), 144-155.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/g4g7.pdf">Seven Staggering Sequences</a>.
%H Maryna S. Viazovska, <a href="http://arxiv.org/abs/1603.04246">The sphere packing problem in dimension 8</a>, arXiv preprint arXiv:1603.04246 [math.NT], 2016.
%H Martin H. Weissman, <a href="http://people.ucsc.edu/~weissman/SAGE13Slides.pdf">Octonions, Cubes, Embeddings</a>, March 2, 2009.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EisensteinSeries.html">Eisenstein Series.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LeechLattice.html">Leech Lattice.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Barnes-WallLattice.html">Barnes-Wall Lattice</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Eisenstein_series">Eisenstein series</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/E8_lattice">E_8 lattice</a>
%H <a href="/index/Ed#Eisen">Index entries for sequences related to Eisenstein series</a>
%H <a href="/index/Ba#BW">Index entries for sequences related to Barnes-Wall lattices</a>
%F 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
%F 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).
%F 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.
%F 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
%F 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
%F 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
%F 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
%F 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
%F Convolution square is A008410. A008411 is convolution of this sequence with A008410.
%F Expansion of Ramanujan's function Q(q^2) = 12 (omega/Pi)^4 g2 (Weierstrass invariant) in powers of q^2.
%F 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
%F 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
%F a(n) = 240*A001158(n) if n>0. - _Michael Somos_, Oct 01 2018
%F Sum_{k=1..n} a(k) ~ 2 * Pi^4 * n^4 / 3. - _Vaclav Kotesovec_, Jan 14 2024
%e G.f. = 1 + 240*x + 2160*x^2 + 6720*x^3 + 17520*x^4 + 30240*x^5 + 60480*x^6 + ...
%e G.f. = 1 + 240*q^2 + 2160*q^4 + 6720*q^6 + 17520*q^8 + 30240*q^10 + 60480*q^12 + ...
%p 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);
%t a[ n_] := If[ n < 1, Boole[n == 0], 240 DivisorSigma[ 3, n]]; (* _Michael Somos_, Jul 11 2011 *)
%t 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 *)
%t 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_ *)
%t 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 *)
%o (PARI) {a(n) = if( n<1, n==0, 240 * sigma(n, 3))};
%o (PARI) {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 */
%o (PARI) 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
%o (Sage) ModularForms(Gamma1(1), 4, prec=30).0 ; # _Michael Somos_, Jun 04 2013
%o (Magma) Basis( ModularForms( Gamma1(1), 4), 29) [1]; /* _Michael Somos_, May 11 2015 */
%o (Magma) L := Lattice("E",8); A<q> := ThetaSeries(L, 57); A; /* _Michael Somos_, Jun 10 2019 */
%o (Python)
%o from sympy import divisor_sigma
%o def a(n): return 1 if n == 0 else 240 * divisor_sigma(n, 3)
%o [a(n) for n in range(51)] # _Indranil Ghosh_, Jul 15 2017
%Y Cf. A046948 (partial sums), A000143, A108091 (eighth root).
%Y Cf. A008410, A008411, A001158.
%Y 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).
%Y Cf. A007331 (theta_2(q)^8 / 256), A000143 (theta_3(q)^8)), A035016 (theta_4(q)^8).
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
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