A028594 Expansion of (theta_3(q) * theta_3(q^7) + theta_2(q) * theta_2(q^7))^2 in powers of q.

1, 4, 12, 16, 28, 24, 48, 4, 60, 52, 72, 48, 112, 56, 12, 96, 124, 72, 156, 80, 168, 16, 144, 96, 240, 124, 168, 160, 28, 120, 288, 128, 252, 192, 216, 24, 364, 152, 240, 224, 360, 168, 48, 176, 336, 312, 288, 192

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Theta series of square of Kleinian lattice Z[ (-1+sqrt(-7))/2 ].

The Gram matrix of the lattice is denoted by A in Parry 1979 on page 163.

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

Examples

			G.f. = 1 + 4*q + 12*q^2 + 16*q^3 + 28*q^4 + 24*q^5 + 48*q^6 + 4*q^7 + 60*q^8 + ...
		

References

• B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag; see p. 467, Entry 5(i).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• W. R. Parry, A negative result on the representation of modular forms by theta series, J. Reine Angew. Math., 310 (1979), 151-170.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A002652, A113957.

Programs

Magma

Basis( ModularForms( Gamma0(7), 2), 48) [1]; /* Michael Somos, Jun 12 2014 */

Mathematica

a[ n_] := If[ n < 1, Boole[ n == 0], 4 Sum[ If[ Mod[ d, 7] > 0, d, 0], {d, Divisors @ n }]]; (* Michael Somos, Jun 12 2014 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^7] + EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^7])^2, {q, 0, n}]; (* Michael Somos, Jun 12 2014 *)

PARI

{a(n) = if( n<1, n==0, 4 * sigma( n / 7^valuation( n, 7)))}; /* Michael Somos, Oct 07 2005 */

PARI

{a(n) = if( n<1, n==0, 2 * qfrep( [2, 1, 0, 0; 1, 4, 0, 0; 0, 0, 2 ,1 ; 0, 0, 1, 4], n, 1)[n])}; /* Michael Somos, Oct 07 2005 */

PARI

{a(n) = if( n<1, n==0, 4 * sumdiv( n, d, d * kronecker( 49, d)))}; /* Michael Somos, Mar 22 2012 */

Sage

ModularForms( Gamma0(7), 2, prec=48).0; # Michael Somos, Jun 12 2014

Formula

Expansion of (phi(q) * phi(q^7) + 4 * q^2 * psi(q^2) * psi(q^14))^2 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Jul 21 2012

Expansion of (7 * P(q^7) - P(q)) / 6 where P() is a Ramanujan Eisenstein Series. - Michael Somos, Mar 22 2012

a(n) = 4 * b(n) where b(n) is multiplicative with b(p^e) = 1, if p=7, b(p^e) = (p^(e+1) - 1) / (p - 1) otherwise.

G.f. is a period 1 Fourier series which satisfies f(-1 / (7 t)) = 7 (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Mar 22 2012

G.f.: (theta_3(q) * theta_3(q^7) + theta_2(q) * theta_2(q^7))^2.

G.f.: 1 + 4 * (Sum_{k>0} Kronecker( 49, k) * k * x^k / (1 - x^k)). - Michael Somos, Mar 22 2012

G.f.: 1 + 4 * (Sum_{k>0} x^k / (1 - x^k)^2 - 7 * x^(7*k) / (1 - x^(7*k))^2). - Michael Somos, Mar 22 2012

Convolution square of A002652. a(n) = 4 * A113957(n) unless n=0. - Michael Somos, Jul 21 2012