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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.

%I A028594 #38 Aug 11 2025 06:57:40
%S A028594 1,4,12,16,28,24,48,4,60,52,72,48,112,56,12,96,124,72,156,80,168,16,
%T A028594 144,96,240,124,168,160,28,120,288,128,252,192,216,24,364,152,240,224,
%U A028594 360,168,48,176,336,312,288,192
%N A028594 Expansion of (theta_3(q) * theta_3(q^7) + theta_2(q) * theta_2(q^7))^2 in powers of q.
%C A028594 Theta series of square of Kleinian lattice Z[ (-1+sqrt(-7))/2 ].
%C A028594 The Gram matrix of the lattice is denoted by A in Parry 1979 on page 163.
%C A028594 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A028594 Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).
%D A028594 B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag; see p. 467, Entry 5(i).
%H A028594 G. C. Greubel, <a href="/A028594/b028594.txt">Table of n, a(n) for n = 0..10000</a>
%H A028594 W. R. Parry, <a href="https://gdz.sub.uni-goettingen.de/id/PPN243919689_0310">A negative result on the representation of modular forms by theta series</a>, J. Reine Angew. Math., 310 (1979), 151-170.
%H A028594 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A028594 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A028594 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
%F A028594 Expansion of (7 * P(q^7) - P(q)) / 6 where P() is a Ramanujan Eisenstein Series. - _Michael Somos_, Mar 22 2012
%F A028594 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.
%F A028594 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
%F A028594 G.f.: (theta_3(q) * theta_3(q^7) + theta_2(q) * theta_2(q^7))^2.
%F A028594 G.f.: 1 + 4 * (Sum_{k>0} Kronecker( 49, k) * k * x^k / (1 - x^k)). - _Michael Somos_, Mar 22 2012
%F A028594 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
%F A028594 Convolution square of A002652. a(n) = 4 * A113957(n) unless n=0. - _Michael Somos_, Jul 21 2012
%e A028594 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 + ...
%t A028594 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 *)
%t A028594 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 *)
%o A028594 (PARI) {a(n) = if( n<1, n==0, 4 * sigma( n / 7^valuation( n, 7)))}; /* _Michael Somos_, Oct 07 2005 */
%o A028594 (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 */
%o A028594 (PARI) {a(n) = if( n<1, n==0, 4 * sumdiv( n, d, d * kronecker( 49, d)))}; /* _Michael Somos_, Mar 22 2012 */
%o A028594 (Sage) ModularForms( Gamma0(7), 2, prec=48).0; # _Michael Somos_, Jun 12 2014
%o A028594 (Magma) Basis( ModularForms( Gamma0(7), 2), 48) [1]; /* _Michael Somos_, Jun 12 2014 */
%Y A028594 Cf. A002652, A113957.
%K A028594 nonn
%O A028594 0,2
%A A028594 _N. J. A. Sloane_