A262160 - OEIS

%I #14 Mar 12 2021 22:24:48

%S 1,-1,1,-2,3,-4,6,-8,11,-15,19,-25,33,-42,53,-68,86,-107,134,-166,205,

%T -253,309,-377,460,-557,672,-811,974,-1166,1394,-1661,1975,-2344,2773,

%U -3275,3863,-4543,5333,-6253,7316,-8544,9964,-11600,13484,-15653,18140

%N Expansion of psi(x^6) / psi(x) in powers of x where psi() 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="/A262160/b262160.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^(-5/8) * eta(q) * eta(q^12)^2 / (eta(q^2)^2 * eta(q^6)) in powers of q.

%F Euler transform of period 12 sequence [ -1, 1, -1, 1, -1, 2, -1, 1, -1, 1, -1, 0, ...].

%F a(n) = (-1)^n * A132217(n).

%F Product_{k>0} (1 - x^(12*k)) * (1 - x^(2*k) + x^(4*k)) / (1 - (-x)^k). - _Michael Somos_, Oct 04 2015

%e G.f. = 1 - x + x^2 - 2*x^3 + 3*x^4 - 4*x^5 + 6*x^6 - 8*x^7 + 11*x^8 + ...

%e G.f. = q^5 - q^13 + q^21 - 2*q^29 + 3*q^37 - 4*q^45 + 6*q^53 - 8*q^61 + ...

%t a[ n_] := SeriesCoefficient[ x^(-5/8) EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, n}];

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

%Y Cf. A132217.

%K sign

%O 0,4

%A _Michael Somos_, Sep 13 2015