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%I #20 Mar 12 2021 22:24:44
%S 1,2,0,0,-5,-14,0,0,2,34,0,0,25,-28,0,0,-28,0,0,0,-46,-14,0,0,49,4,0,
%T 0,68,82,0,0,0,-62,0,0,-142,50,0,0,-11,-158,0,0,146,0,0,0,-94,70,0,0,
%U 0,178,0,0,98,0,0,0,75,-92,0,0,-28,-62,0,0,-238,-206,0
%N Expansion of phi(x) * psi(x^4) * phi(-x^4)^4 in powers of x where phi(), psi() are Ramanujan theta functions.
%C Number 48 of the 74 eta-quotients listed in Table I of Martin (1996).
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). - _Michael Somos_, Mar 14 2012
%H G. C. Greubel, <a href="/A128711/b128711.txt">Table of n, a(n) for n = 0..1000</a>
%H Y. 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 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</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^(-1/2) * (eta(q^2) * eta(q^4))^5 / (eta(q) * eta(q^8))^2 in powers of q. - _Michael Somos_, Mar 14 2012
%F Euler transform of period 8 sequence [ 2, -3, 2, -8, 2, -3, 2, -6, ...].
%F a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^e if p == 5, 7 (mod 8), b(p^e) = b(p) * b(p^(e-1)) - p^2 * b(p^(e-2)) if p == 1, 3 (mod 8) where b(p) = 2*(2*x^2 - p) * (-1)^((p-1)/2) and p = x^2 + 2*y^2.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(15/2) (t/i)^3 f(t) where q = exp(2 Pi i t).
%F G.f.: Product_{k>0} (1 - x^k)^6 * (1 + x^k)^8 * (1 + x^(2*k))^3 / (1 + x^(4*k))^2.
%F a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A128712(n). a(4*n + 1) = 2 * A128713(n).
%e G.f. = 1 + 2*x - 5*x^4 - 14*x^5 + 2*x^8 + 34*x^9 + 25*x^12 - 28*x^13 + ...
%e G.f. = q + 2*q^3 - 5*q^9 - 14*q^11 + 2*q^17 + 34*q^19 + 25*q^25 - 28*q^27 + ...
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2] QPochhammer[ x^4])^5 / (QPochhammer[ x] QPochhammer[ x^8])^2, {x, 0, n}]; (* _Michael Somos_, Jul 09 2015 *)
%o (PARI) {a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p%8>4, if( e%2, 0, p^e), for( i=1, sqrtint(p\2), if( issquare(p - 2*i^2, &x), break)); a0=1; a1=y=2*(2*x^2 - p) * (-1)^((p-1)/2); for( i=2, e, x = y*a1 - p^2*a0; a0=a1; a1=x); a1)))};
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^4 + A))^5 / (eta(x + A) * eta(x^8 + A))^2, n))};
%Y Cf. A128712, A128713.
%K sign
%O 0,2
%A _Michael Somos_, Mar 24 2007
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