A279255 - OEIS

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

%S 1,1,0,0,0,1,0,0,0,-1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,-1,0,0,

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

%U 0,4,0,0,0,2,0,0,-1,-5,0,0,0,-2,0,0,0,5,0

%N Expansion of chi(x) * chi(-x^3) * chi(-x^8) * chi(-x^24) in powers of x where chi() is a Ramanujan 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="/A279255/b279255.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 Euler transform of period 48 sequence [ 1, -1, 0, 0, 1, -1, 1, -1, 0, -1, 1, 0, 1, -1, 0, 0, 1, -1, 1, 0, 0, -1, 1, -2, 1, -1, 0, 0, 1, -1, 1, 0, 0, -1, 1, 0, 1, -1, 0, -1, 1, -1, 1, 0, 0, -1, 1, 0, ...].

%F a(4*n + 2) = a(4*n + 3) = a(6*n + 2) = a(6*n + 4) = 0.

%F a(4*n + 1) = a(12*n) = A029838(n).

%e G.f. = 1 + x + x^5 - x^9 + x^12 + x^17 - x^24 - x^25 - x^29 + 2*x^33 + ...

%e G.f. = q^-3 + q^-1 + q^7 - q^15 + q^21 + q^31 - q^45 - q^47 - q^55 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ x^3, x^6] QPochhammer[ x^8, x^16] QPochhammer[ x^24, x^48], {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^8 + A) * eta(x^24 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^16 + A) * eta(x^48 + A)), n))};

%Y Cf. A029838.

%K sign

%O 0,34

%A _Michael Somos_, Dec 08 2016