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A028594 Expansion of (theta_3(q) * theta_3(q^7) + theta_2(q) * theta_2(q^7))^2 in powers of q. 7
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 (list; graph; refs; listen; history; text; internal format)
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
W. R. Parry, A negative result on the representation of modular forms by theta series, J. Reine Angew. Math., 310 (1979), 151-170.
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
Sequence in context: A090818 A075191 A320922 * A239050 A152680 A270248
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified April 24 15:42 EDT 2024. Contains 371960 sequences. (Running on oeis4.)