A260941 - OEIS

%I #10 Mar 12 2021 22:24:48

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

%T 0,0,0,-8,0,1,0,0,2,0,0,2,0,0,3,0,0,0,0,0,2,-4,0,2,-6,0,2,0,0,4,-4,0,

%U 0,-4,0,1,0,0,4,0,0,2,0,0,2,0,0,1,-4,0,0,-4

%N Expansion of phi(-x) * phi(x^6) / chi(-x^3) in powers of x where phi(), chi() are Ramanujan theta functions.

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

%H G. C. Greubel, <a href="/A260941/b260941.txt">Table of n, a(n) for n = 0..2500</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 q^(-1/8) * eta(q)^2 * eta(q^12)^5 / (eta(q^2) * eta(q^3) * eta(q^6) * eta(q^24)^2) in powers of q.

%F Euler transform of period 24 sequence [ -2, -1, -1, -1, -2, 1, -2, -1, -1, -1, -2, -4, -2, -1, -1, -1, -2, 1, -2, -1, -1, -1, -2, -2, ...].

%F a(3*n) = A131961(n). a(3*n + 1) = -2 * A112608(n). a(3*n + 2) = 0.

%e G.f. = 1 - 2*x + x^3 + 3*x^6 - 4*x^7 + 2*x^9 - 2*x^10 + 2*x^12 + x^15 + ...

%e G.f. = q - 2*q^9 + q^25 + 3*q^49 - 4*q^57 + 2*q^73 - 2*q^81 + 2*q^97 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 3, 0, x^6] QPochhammer[ -x^3, x^3], {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^12 + A)^5 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A) * eta(x^24 + A)^2), n))};

%o (PARI) q='q+O('q^99); Vec(eta(q)^2*eta(q^12)^5/(eta(q^2)*eta(q^3)*eta(q^6)*eta(q^24)^2)) \\ _Altug Alkan_, Aug 01 2018

%Y Cf. A112608, A131961.

%K sign

%O 0,2

%A _Michael Somos_, Aug 04 2015