A256279 - OEIS

%I #12 Feb 16 2025 08:33:25

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

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

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

%N Expansion of psi(q) * chi(-q^3) * phi(-q^9) in powers of q where phi(), psi(), 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="/A256279/b256279.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="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of eta(q^2)^2 * eta(q^3) * eta(q^9)^2 / (eta(q) * eta(q^6) * eta(q^18)) in powers of q.

%F Euler transform of period 18 sequence [ 1, -1, 0, -1, 1, -1, 1, -1, -2, -1, 1, -1, 1, -1, 0, -1, 1, -2, ...].

%F a(n) = (-1)^n * A256269(n). a(4*n) = A256269(n).

%F a(3*n + 2) = a(4*n + 3) = 0. a(3*n + 1) = A258277(n). a(6*n + 4) = - A122856(n). a(12*n + 1) = A002175(n). a(12*n + 4) = - A122865(n).

%e G.f. = 1 + q - q^4 - 4*q^9 - 2*q^10 + 2*q^13 - q^16 + 4*q^18 + 3*q^25 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)] / (2 q^(1/8)) QPochhammer[ q^3, q^6] EllipticTheta[ 4, 0, q^9], {q, 0, n}];

%o (PARI) {a(n) = if( n<1, n==0, (-1)^(n\3) * (n%3<2) * sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))};

%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^9 + A)^2 / (eta(x + A) * eta(x^6 + A) * eta(x^18 + A)), n))};

%Y Cf. A002175, A122856, A122865, A256269, A258277.

%K sign,changed

%O 0,10

%A _Michael Somos_, Jun 02 2015