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A120030 Expansion of theta_4(q)^2*theta_4(q^2)^4 in powers of q.

%I A120030 #34 Aug 06 2025 10:08:23
%S A120030 1,-4,-4,32,-4,-104,32,192,-4,-292,-104,480,32,-680,192,832,-4,-1160,
%T A120030 -292,1440,-104,-1536,480,2112,32,-2604,-680,2624,192,-3368,832,3840,
%U A120030 -4,-3840,-1160,4992,-292,-5480,1440,5440,-104,-6728,-1536,7392,480,-7592,2112,8832,32,-9412,-2604
%N A120030 Expansion of theta_4(q)^2*theta_4(q^2)^4 in powers of q.
%C A120030 Number 8 of the 74 eta-quotients listed in Table I of Martin (1996).
%C A120030 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A120030 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.7).
%H A120030 G. C. Greubel, <a href="/A120030/b120030.txt">Table of n, a(n) for n = 0..10000</a>
%H A120030 David Broadhurst and Daniele Dorigoni, <a href="https://arxiv.org/abs/2507.21352">Resurgent Lambert series with characters</a>, arXiv:2507.21352 [math.NT], 2025. See p. 37.
%H A120030 Frazer Jarvis and Helena A. Verrill, <a href="https://doi.org/10.1007/s11139-009-9218-5">Supercongruences for the Catalan-Larcombe-French numbers</a>, Ramanujan J (22) (2010) 171.
%H A120030 Yves Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H A120030 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%H A120030 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A120030 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A120030 Expansion of eta(q)^4 * eta(q^2)^6 / eta(q^4)^4 in powers of q.
%F A120030 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^2 * (u - v)^2 - 4 * u*w * (v - w) * (u - 2*v).
%F A120030 Euler transform of period 4 sequence [ -4, -10, -4, -6, ...].
%F A120030 G.f.: 1 - 4 * Sum_{k>0} A056594(k-1) * k^2 * x^k / (1 - x^k).
%F A120030 Expansion of phi(-q)^2 * phi(-q^2)^4 in powers of q where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Aug 15 2007
%F A120030 G.f.: (Sum_{k in Z} (-1)^k * x^k^2)^2 * (Sum_{k in Z} (-1)^k * x^(2*k^2))^4.
%F A120030 G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 128 (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A050470.
%F A120030 a(n) = -4 * A002173(n) unless n=0.
%F A120030 Convolution of A000141 and A128692.
%e A120030 G.f. = 1 - 4*q - 4*q^2 + 32*q^3 - 4*q^4 - 104*q^5 + 32*q^6 + 192*q^7 - 4*q^8 + ...
%t A120030 a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^3 (QPochhammer[ q] / QPochhammer[ q^4])^2)^2, {q, 0, n}]; (* _Michael Somos_, May 24 2015 *)
%t A120030 a[ n_] := If[ n < 1, Boole[n == 0], -4 DivisorSum[ n, #^2 KroneckerSymbol[ -4, #] &]]; (* _Michael Somos_, May 24 2015 *)
%o A120030 (PARI) {a(n) = if( n<1, n==0, -4 * sumdiv( n, d, d^2 * kronecker( -4, d)))};
%o A120030 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^2 + A)^6 / eta(x^4 + A)^4, n))};
%o A120030 (Magma) A := Basis( ModularForms( Gamma1(4), 3), 51); A[1] - 4*A[2]; /* _Michael Somos_, May 24 2015 */
%Y A120030 Cf. A000141, A002173, A050470, A128692.
%K A120030 sign
%O A120030 0,2
%A A120030 _Michael Somos_, Jun 05 2006