A255257 - OEIS

%I #9 Mar 12 2021 22:24:47

%S 1,1,-4,-3,4,0,1,4,0,4,-3,-4,-4,-8,8,1,-4,0,0,4,0,5,4,8,-4,-4,4,-8,-3,

%T -4,4,-4,0,0,-8,4,1,0,-8,0,4,8,8,8,0,1,0,-8,8,-4,-4,-8,12,4,-12,1,-4,

%U 0,0,-4,-8,4,-8,0,0,-8,1,12,8,8,0,-8,8,0,8,4,0

%N Expansion of psi(x) * phi(-x^2)^2 in powers of x where phi(), psi() 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="/A255257/b255257.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 q^(-1/8) * eta(q^2)^6 / (eta(q) * eta(q4)^2) in powers of q.

%F Euler transform of period 4 sequence [ 1, -5, 1, -3, ...].

%F a(n) = A034950(4*n).

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

%e G.f. = q + q^9 - 4*q^17 - 3*q^25 + 4*q^33 + q^49 + 4*q^57 + 4*q^73 + ...

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

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

%Y Cf. A034950.

%K sign

%O 0,3

%A _Michael Somos_, Feb 19 2015