A028597 - OEIS

%I #20 Mar 12 2021 22:24:41

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

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

%U 4,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,4,0,0,2,0,0,0,0

%N Expansion of theta_3(z) * theta_3(8*z) + theta_2(z) * theta_2(8*z) in powers of q^(1/4).

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

%H G. C. Greubel, <a href="/A028597/b028597.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>

%F Expansion of phi(q) * phi(q^32) - 2 * q * f(-q^8) * f(-q^16) = phi(q^4) * phi(q^32) + 4 * q^9 * phi(q^8) * psi(q^64) in powers of q where phi(), psi(), f() are Ramanujan theta functions. - _Michael Somos_, Jun 24 2011

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (128 t)) = 128^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - _Michael Somos_, Jun 24 2011

%e G.f. = 1 + 2*q^4 + 4*q^9 + 2*q^16 + 4*q^17 + 2*q^32 + 4*q^33 + 6*q^36 + 4*q^48 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^32] - 2 q QPochhammer[ q^8] QPochhammer[ q^16], {q, 0, n}]; (* _Michael Somos_, Aug 26 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum( k=1, sqrtint( n), 2 * x^k^2, 1 + A) * sum( k=1, sqrtint( n\32), 2 * x^(32*k^2), 1 + A) - 2 * x * eta(x^8 + A) * eta(x^16 + A), n))}; /* _Michael Somos_, Jun 24 2011 */

%K nonn

%O 0,5

%A _N. J. A. Sloane_, Dec 11 1999