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A170772 Expansion of ( phi(q) * phi(q^63) + phi(-q) * phi(-q^63) + 4 * q^16 * psi(q^2) * psi(q^126) ) ^ 2 in powers of q^2 where phi(), psi() are Ramanujan theta functions. 4

%I

%S 4,0,16,0,16,0,0,0,32,16,64,48,0,32,16,0,96,64,48,64,96,0,80,48,0,64,

%T 128,64,16,80,0,96,144,0,128,0,112,96,128,0,224,160,0,128,240,96,208,

%U 160,0,0,304,0,256,112,192,224,32,0,240,128,0,192,224,16,336

%N Expansion of ( phi(q) * phi(q^63) + phi(-q) * phi(-q^63) + 4 * q^16 * psi(q^2) * psi(q^126) ) ^ 2 in powers of q^2 where phi(), psi() are Ramanujan theta functions.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%D Bruce C. Berndt, Ramanujan's Notebooks, Part IV, Springer-Verlag, 1984; see Entry 27, pp. 170-171. This is the right side of (27.1).

%H G. C. Greubel, <a href="/A170772/b170772.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 Michael Somos, <a href="/A170773/a170773.txt">Notes on Entry 27 of Chapter 25 of Ramanujan's Notebooks, Part IV</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%e G.f. = 4 + 16*x^2 + 16*x^4 + 32*x^8 + 16*x^9 + 64*x^10 + 48*x^11 + ...

%e G.f. = 4 + 16*q^4 + 16*q^8 + 32*q^16 + 16*q^18 + 64*q^20 + 48*q^22 + 32*q^26 + 16*q^28 + ...

%t QP = QPochhammer; f[q_] := QPochhammer[-q, -q]; p[q_] := EllipticTheta[3, 0, q]; u[q_] := (1/2)*q^(-1/8)*EllipticTheta[2, 0, Sqrt[q]]; a[n_] := SeriesCoefficient[(p[q]*p[q^63] + p[-q]*p[-q^63] + 4*q^16*u[q^2]*u[q^(126)])^2, {q, 0, n}]; Table[a[n], {n, 0, 100}][[ ;; ;; 2]] (* _G. C. Greubel_, Dec 05 2017 *)

%Y Cf. A170770, A170771, A170773.

%K nonn

%O 0,1

%A _Michael Somos_, Dec 10 2009

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Last modified June 16 21:15 EDT 2021. Contains 345080 sequences. (Running on oeis4.)