A170770 - OEIS

%I #21 Feb 16 2025 08:33:11

%S 1,0,2,0,0,0,0,0,4,2,0,2,0,0,2,0,0,0,4,0,0,0,0,2,0,0,0,0,0,2,0,0,6,0,

%T 0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,0,2,0,0,2,0,0,4,0,0,0,0,0,0,2,0,0,0,0,

%U 0,0,0,2,8,0,4,0,0,2,0,0,0,2,0,0,0,0,4,0,0,0,0,0,6,0,0,0,0,0,2

%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 square root of the right side of (27.1), divided by 2.

%H G. C. Greubel, <a href="/A170770/b170770.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="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F G.f.: ( phi(q^7) * phi(q^9) + phi(-q^7) * phi(-q^9) + 4 * q^4 * psi(q^14) * psi(q^18) ) / 2 in powers of q^2 where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, May 24 2018

%e G.f. = 1 + 2*x^2 + 4*x^8 + 2*x^9 + 2*x^11 + 2*x^14 + 4*x^18 + 2*x^23 + ...

%e G.f. = 1 + 2*q^4 + 4*q^16 + 2*q^18 + 2*q^22 + 2*q^28 + 4*q^36 + 2*q^46 + 2*q^58 + ...

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

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^7] EllipticTheta[ 3, 0, q^9] + 1/2 EllipticTheta[ 2, 0, q^7] EllipticTheta[ 2, 0, q^9], {q, 0, 2 n}, Assumptions -> q>0]; (* _Michael Somos_, May 24 2018 *)

%Y Cf. A170771, A170772, A170773.

%K nonn,changed

%O 0,3

%A _Michael Somos_, Dec 10 2009