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A228745 Expansion of (phi(q)^4 + 7 * phi(-q)^4) / 8 in powers of q where phi() is a Ramanujan theta function.

%I A228745 #16 Feb 16 2025 08:33:20
%S A228745 1,-6,24,-24,24,-36,96,-48,24,-78,144,-72,96,-84,192,-144,24,-108,312,
%T A228745 -120,144,-192,288,-144,96,-186,336,-240,192,-180,576,-192,24,-288,
%U A228745 432,-288,312,-228,480,-336,144,-252,768,-264,288,-468,576,-288,96,-342,744
%N A228745 Expansion of (phi(q)^4 + 7 * phi(-q)^4) / 8 in powers of q where phi() is a Ramanujan theta function.
%C A228745 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A228745 G. C. Greubel, <a href="/A228745/b228745.txt">Table of n, a(n) for n = 0..2500</a>
%H A228745 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A228745 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A228745 a(n) = -6 * b(n) where b() is multiplicative with a(0) = 1, b(2^e) = -4 if e>1, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)), if p>2.
%F A228745 G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 1/2 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A228746.
%F A228745 G.f.: ( (Sum_{k in Z} x^k^2)^4 + 7 * (Sum_{k in Z} (-x)^k^2)^4 ) / 8.
%F A228745 a(2*n) = A004011(n). a(2*n + 1) = -6 * A008438(n).
%F A228745 Convolution with A005875 is A003781.
%e A228745 G.f. = 1 - 6*q + 24*q^2 - 24*q^3 + 24*q^4 - 36*q^5 + 96*q^6 - 48*q^7 + 24*q^8 + ...
%t A228745 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^4 + 7 EllipticTheta[ 4, 0, q]^4) / 8, {q, 0, n}];
%o A228745 (PARI) {a(n) = my(A); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( (A^4 + 7 * subst(A, x, -x)^4) / 8, n))};
%o A228745 (Magma) A := Basis( ModularForms( Gamma0(4), 2), 51); A[1] - 6*A[2]; /* _Michael Somos_, Aug 21 2014 */
%Y A228745 Cf. A003781, A004011, A005875, A008438, A228746.
%K A228745 sign
%O A228745 0,2
%A A228745 _Michael Somos_, Sep 02 2013