A226861 - OEIS

%I #11 Mar 12 2021 22:24:47

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

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

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

%N Expansion of phi(x) * f(-x^3) in powers of x where phi(), f() 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="/A226861/b226861.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)^5 * eta(q^3) / (eta(q) * eta(q^4))^2 in powers of q.

%F Euler transform of period 12 sequence [2, -3, 1, -1, 2, -4, 2, -1, 1, -3, 2, -2, ...].

%F G.f.: (Sum_{k in Z} x^(k^2)) * Product_{k>0} (1 - x^(3*k)).

%F a(3*n + 2) = 0. a(3*n) = A226289(n).

%e G.f. = 1 + 2*x - x^3 - x^6 - 4*x^7 + 2*x^9 - 2*x^10 - 2*x^12 - x^15 + 4*x^16 + ...

%e G.f. = q + 2*q^9 - q^25 - q^49 - 4*q^57 + 2*q^73 - 2*q^81 - 2*q^97 - q^121 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] QPochhammer[ q^3], {q, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A) / (eta(x + A) * eta(x^4 + A))^2, n))};

%Y Cf. A226289.

%K sign

%O 0,2

%A _Michael Somos_, Jun 20 2013