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%I #16 Mar 12 2021 22:24:46
%S 1,0,4,0,4,0,0,0,8,4,16,12,0,8,4,0,24,16,12,16,24,0,20,12,0,16,32,16,
%T 4,20,0,24,36,0,32,0,28,24,32,0,56,40,0,32,60,24,52,40,0,0,76,0,64,28,
%U 48,56,8,0,60,32,0,48,56,4,84,48,0,48,88,0,16,36,60,40,80,0
%N Expansion of ( phi(q) * phi(q^7) * phi(q^9) * phi(q^63) + phi(-q) * phi(-q^7) * phi(-q^9) * phi(-q^63) + 4 * q^4 * f(-q^6)^2 * f(-q^42)^2 + 16 * q^20 * psi(q^2) * psi(q^14) * psi(q^18) * psi(q^126) ) / 2 in powers of q^2 where phi(), psi(), and f() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C This is the convolution square of A170770.
%D Bruce C. Berndt, Ramanujan's Notebooks, Part IV, Springer-Verlag, 1984; see Entry 27, pp. 170-171. This is the corrected left side of (27.1), divided by 4.
%H G. C. Greubel, <a href="/A170773/b170773.txt">Table of n, a(n) for n = 0..1000</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>
%H Michael Somos, <a href="/A170773/a170773.txt">Notes on Entry 27 of Chapter 25 of Ramanujan's Notebooks, Part IV</a>
%e G.f. = 1 + 4*x^2 + 4*x^4 + 8*x^8 + 4*x^9 + 16*x^10 + 12*x^11 + 8*x^13 + ...
%e G.f. = 1 + 4*q^4 + 4*q^8 + 8*q^16 + 4*q^18 + 16*q^20 + 12*q^22 + 8*q^26 + 4*q^28 + ...
%t f[q_] := QPochhammer[-q, -q]; p[q_] := EllipticTheta[3, 0, q]; u[q_] := (1/2)*q^(-1/8)*EllipticTheta[2, 0, Sqrt[q]]; A170773[n_] := SeriesCoefficient[(1/2)*(p[q]*p[q^7]*p[q^9]*p[q^63] + p[-q]*p[-q^7]*p[-q^9]*p[-q^63] + 4*q^4*f[-q^6]^2*f[-q^42]^2 + 16*q^20*u[q^2]*u[q^14]*u[q^18]*u[q^(126)]), {q, 0, n}]; Table[A170773[n], {n, 0, 100}][[ ;; ;; 2]] (* _G. C. Greubel_, Dec 03 2017 *)
%Y Cf. A170770-A170772.
%K nonn
%O 0,3
%A _Michael Somos_, Dec 10 2009
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