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A261277 Expansion of q^(-1/2) * (eta(q^3)^8 + 4 * eta(q^6)^8)^(1/2) in powers of q.

%I A261277 #10 Sep 08 2022 08:46:13
%S A261277 1,2,-2,0,-2,4,-2,-4,2,-4,0,-8,-1,2,6,8,8,0,6,-4,-6,4,4,0,-7,4,-2,-8,
%T A261277 -8,4,-2,0,4,-4,-16,8,10,-2,0,-8,-2,-4,-4,12,-6,0,16,8,2,-8,-18,16,0,
%U A261277 -12,-2,12,18,16,4,0,5,-12,12,-8,8,-4,0,-4,-6,-12,0,-8
%N A261277 Expansion of q^(-1/2) * (eta(q^3)^8 + 4 * eta(q^6)^8)^(1/2) in powers of q.
%C A261277 It seems that a( (p - 1)/2 ) = 0 if and only if p is in A167860.
%H A261277 G. C. Greubel, <a href="/A261277/b261277.txt">Table of n, a(n) for n = 0..2500</a>
%F A261277 a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = -2 * (-1)^e if e>0, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) if p>3.
%F A261277 G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 72 (t/i)^2 f(t) where q = exp(2 Pi i t).
%F A261277 Convolution square is A261278.
%e A261277 G.f. = 1 + 2*x - 2*x^2 - 2*x^4 + 4*x^5 - 2*x^6 - 4*x^7 + 2*x^8 - 4*x^9 - 8*x^11 + ...
%e A261277 G.f. = q + 2*q^3 - 2*q^5 - 2*q^9 + 4*q^11 - 2*q^13 - 4*q^15 + 2*q^17 - 4*q^19 + ...
%t A261277 a[ n_] := SeriesCoefficient[ Sqrt[ QPochhammer[ x^3]^8 + 4 x QPochhammer[ x^6]^8], {x, 0, n}];
%o A261277 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sqrt( eta(x^3 + A)^8 + 4 * x * eta(x^6 + A)^8 ), n))};
%o A261277 (Magma) A := Basis( CuspForms( Gamma0(72), 2), 142); A[1] + 2*A[3] - 2*A[4];
%Y A261277 Cf. A167860, A261278.
%K A261277 sign
%O A261277 0,2
%A A261277 _Michael Somos_, Aug 14 2015