A263548 - OEIS

%I #9 Mar 12 2021 22:24:48

%S 1,2,1,2,2,0,2,0,0,2,1,4,0,0,2,0,3,2,2,2,2,0,0,0,0,4,2,2,0,0,0,0,3,2,

%T 0,2,4,0,2,0,0,4,1,2,0,0,4,0,2,2,0,4,2,0,0,0,0,2,2,2,0,0,0,0,2,2,3,4,

%U 2,0,2,0,0,0,2,2,0,0,2,0,3,6,2,2,2,0,0

%N Expansion of f(x, x) * f(x^2, x^10) in powers of x where f(, ) is Ramanujan's general theta function.

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

%H G. C. Greubel, <a href="/A263548/b263548.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(x) * chi(x^2) * psi(-x^6) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

%F Expansion of q^(-2/3) * eta(q^2)^4 * eta(q^6) * eta(q^24) / (eta(q)^2 * eta(q^8) * eta(q^12)) in powers of q.

%F Euler transform of period 24 sequence [ 2, -2, 2, -2, 2, -3, 2, -1, 2, -2, 2, -2, 2, -2, 2, -1, 2, -3, 2, -2, 2, -2, 2, -2, ...].

%F a(n) = A000377(3*n + 2) = A192013(3*n + 2) = A115660(6*n + 4).

%e G.f. = 1 + 2*x + x^2 + 2*x^3 + 2*x^4 + 2*x^6 + 2*x^9 + x^10 + 4*x^11 + ...

%e G.f. = q^2 + 2*q^5 + q^8 + 2*q^11 + 2*q^14 + 2*q^20 + 2*q^29 + q^32 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^2, x^12] QPochhammer[ -x^10, x^12] QPochhammer[ x^12], {x, 0, n}];

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^2, x^4] EllipticTheta[ 2, Pi/4, x^3] / (2^(1/2) x^(3/4)), {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^6 + A) * eta(x^24 + A) / (eta(x + A)^2 * eta(x^8 + A) * eta(x^12 + A)), n))};

%Y Cf. A000377, A115660, A192013.

%K nonn

%O 0,2

%A _Michael Somos_, Oct 20 2015