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

0,2

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).

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

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

Example

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 + ...

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 *)

Prog

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

Crossrefs

Cf. A002652, A113957.

Sequence in context: A090818 A075191 A320922 * A239050 A152680 A270248

Adjacent sequences: A028591 A028592 A028593 * A028595 A028596 A028597

Keyword

nonn

Author

N. J. A. Sloane

Status

approved