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%I #18 Mar 12 2021 22:24:46
%S 1,-14,81,-238,322,0,-429,82,0,2162,-3038,-1134,2401,2482,0,-6958,
%T 3332,0,1442,0,6561,-4508,-9758,0,-1918,18802,0,9362,-24638,-19278,
%U 14641,14756,0,0,6562,0,-1148,-33998,26082,-20398,0,0,28083,49042,0,-64078,-30268
%N Expansion of (psi(x) * phi(-x)^4)^2 in powers of x where phi(), psi() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C This is a member of an infinite family of integer weight level 8 modular forms. g_1 = A008441, g_2 = A002171, g_3 = A000729, g_4 = A215601, g_5 = A215472.
%H G. C. Greubel, <a href="/A215472/b215472.txt">Table of n, a(n) for n = 0..1000</a>
%H Shobhit, <a href="http://math.stackexchange.com/questions/483708/">How to prove that eta(q^4)^14/eta(q^8)^4 = 4eta(q^2)^4eta(q^4)^2eta(q^8)^4 + eta(q)^4eta(q^2)^2eta(q^4)^4?</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of q^(-1/4) * eta(q)^14 / eta(q^2)^4 in powers of q.
%F Expansion of q^(-1/4) * ( eta(q)^4 * eta(q^2)^2 * eta(q^4)^4 + 4 * eta(q^2)^4 * eta(q^4)^2 * eta(q^8)^4 ) in powers of q. - _Michael Somos_, Sep 05 2013
%F Euler transform of period 2 sequence [ -14, -10, ...].
%F a(n) = b(4*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * p^(2*e) if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) otherwise.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 128 (t/i)^5 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A030212.
%F a(n) = (-1)^n * A209942(n). a(9*n + 5) = a(9*n + 8) = 0. a(9*n + 2) = 81 * a(n).
%F a(n) = A030212(4*n + 1). - _Michael Somos_, Sep 05 2013
%e 1 - 14*x + 81*x^2 - 238*x^3 + 322*x^4 - 429*x^6 + 82*x^7 + 2162*x^9 + ...
%e q - 14*q^5 + 81*q^9 - 238*q^13 + 322*q^17 - 429*q^25 + 82*q^29 + 2162*q^37 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x]^14 / QPochhammer[ x^2]^4, {x, 0, n}] (* _Michael Somos_, Sep 05 2013 *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x + A)^7 / eta(x^2 + A)^2 )^2, n))}
%Y Cf. A000729, A002171, A008441, A030212, A209942, A215601.
%K sign
%O 0,2
%A _Michael Somos_, Aug 12 2012
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