A002653 Expansion of (theta_3(z)*theta_3(7z)+theta_2(z)*theta_2(7z))^3.

1, 6, 24, 56, 114, 168, 280, 294, 444, 390, 840, 636, 1176, 1176, 1512, 1008, 1782, 2016, 1896, 2520, 3528, 2408, 3216, 2796, 4760, 3174, 5880, 4592, 6258, 4380, 5040, 6720, 7200, 6832, 10080, 7224, 8082, 7164, 12600, 7056, 14280, 11760, 12040, 9756

Offset

0,2

Comments

Theta series of Kleinian lattice (Z[ (-1+sqrt(-7))/2 ])^3 in 3 complex (or 6 real) dimensions.

References

N. Elkies, The Klein quartic in number theory, pp. 51-101 of S. Levy, ed., The Eightfold Way, Cambridge Univ. Press, 1999. MR1722413 (2001a:11103)

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 53.

Links

Robert Israel, Table of n, a(n) for n = 0..10000

H. H. Chan and Y. L. Ong, On Eisenstein series and Sum_{m,n} q^(m^2+mn+2n^2), Proc. Amer. Math. Soc. 127 (1999), no. 6, 1735-1744, See page 1737. MR1600120 (99i:11029).

N. Elkies, The Klein quartic in number theory

Formula

G.f.: (theta_3(z)*theta_3(7*z) + theta_2(z)*theta_2(7*z))^3.

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) is a homogeneous degree 6 polynomial with 28 terms. - Michael Somos, Jun 03 2005

Expansion of (eta(q)^8 + 13 * eta(q)^4 * eta(q^7)^4 + 49 * eta(q^7)^8) / ( eta(q) * eta(q^7) ) in power of q. - Michael Somos, Mar 11 2008

G.f. is a period 1 Fourier series which satisfies f(-1 / (7*t)) = 7^(3/2)*(t/i)^3*f(t) where q = exp(2*Pi*i*t).

Example

G.f. = 1 + 6*q + 24*q^2 + 56*q^3 + 114*q^4 + 168*q^5 + 280*q^6 + 294*q^7 + ...

Maple

g:= (JacobiTheta3(0, z) * JacobiTheta3(0, z^7) + JacobiTheta2(0, z) * JacobiTheta2(0, z^7))^3:
S:= series(g, z, 101):
seq(coeff(S, z, i), i=0..100); # Robert Israel, Aug 12 2020

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^7] + EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^7])^3, {q, 0, n}]; (* Michael Somos, Nov 09 2014 *)

Prog

(PARI) {a(n) = local(A, t2, t3); if( n<1, n==0, A = x * O(x^n); t2 = sum(k=1, (sqrtint(4*n + 1) + 1)\2, 2*x^(k*k - k), A); t3 = sum(k=1, sqrtint(n), 2*x^(k*k), 1 + A); A = x * O(x^(n\7)); polcoeff( (t3 * subst(t3 + A, x, x^7) + x^2 * t2 * subst(t2 + A, x, x^7))^3, n))}; /* Michael Somos, Jun 03 2005 */
(PARI) {a(n) = local(A, t1, t7); if( n<0, 0, A = x * O(x^n); t1 = eta(x + A)^4; t7 = eta(x^7 + A)^4; polcoeff( (t1^2 + 13 * x * t1 * t7 + 49 * x^2 * t7^2) / (t1 * t7)^(1/4), n))}; /* Michael Somos, Mar 11 2008 */
(Sage) A = ModularForms( Gamma1(7), 3, prec=60) . basis(); (3*A[0] + 4*A[1] + 21*A[2] + 105*A[3] + 224*A[4] + 441*A[5] + 672*A[6])/4 # _Michael Somos, May 25 2014
(Magma) A := Basis( ModularForms( Gamma1(7), 3), 44); A[1] + 6*A[2] + 24*A[3] + 56*A[4] +114*A[5] + 168*A[6] + 280*A[7];  /* Michael Somos, Nov 09 2014 */

Crossrefs

Cf. A002652.

Sequence in context: A274205 A009943 A028595 * A212904 A264790 A201598

Adjacent sequences: A002650 A002651 A002652 * A002654 A002655 A002656

Keyword

nonn,look

Author

N. J. A. Sloane

Status

approved