A262929 - OEIS

%I #18 Mar 12 2021 22:24:48

%S 1,0,0,-2,-1,0,0,2,1,0,0,-2,0,0,0,4,1,0,0,-6,-2,0,0,8,1,0,0,-12,-1,0,

%T 0,16,2,0,0,-22,-3,0,0,30,2,0,0,-38,-1,0,0,50,4,0,0,-66,-5,0,0,84,3,0,

%U 0,-106,-3,0,0,136,6,0,0,-172,-8,0,0,214,5,0,0

%N Expansion of phi(-x^3) / psi(x^4) 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="/A262929/b262929.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^(1/2) * eta(q^3)^2 * eta(q^4) / (eta(q^6) * eta(q^8)^2) in powers of q.

%F Euler transform of period 24 sequence [0, 0, -2, -1, 0, -1, 0, 1, -2, 0, 0, -2, 0, 0, -2, 1, 0, -1, 0, -1, -2, 0, 0, 0, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = (32/3)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A261877.

%F a(4*n) = A143066(n). a(4*n + 1) = a(4*n + 2) = 0. a(4*n + 3) = -2 * A262160(n).

%F a(12*n) = A262150(n). a(12*n + 3) = -2*A262151(n). a(12*n + 4) = -A262152(n). a(12*n + 7) = 2*A262156(n). a(12*n + 8) = A262157(n). a(12*n + 11) = -2*A262158(n). - _Michael Somos_, Apr 03 2016

%F Convolution inverse is A261877. - _Michael Somos_, Oct 22 2017

%e G.f. = 1 - 2*x^3 - x^4 + 2*x^7 + x^8 - 2*x^11 + 4*x^15 + x^16 + ...

%e G.f. = q^-1 - 2*q^5 - q^7 + 2*q^13 + q^15 - 2*q^21 + 4*q^29 + q^31 + ...

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

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

%o (PARI) q='q+O('q^99); Vec(eta(q^3)^2*eta(q^4)/(eta(q^6)*eta(q^8)^2)) \\ _Altug Alkan_, Jul 31 2018

%Y Cf. A143066, A261877, A262150, A262151, A262152, A262156, A262157, A262158, A262160.

%K sign

%O 0,4

%A _Michael Somos_, Oct 04 2015