A246835 - OEIS

%I #10 Mar 12 2021 22:24:47

%S 1,-2,3,-6,4,-4,7,-2,8,-10,4,-10,9,-6,8,-10,4,-8,16,-8,9,-12,8,-12,20,

%T -6,8,-10,8,-18,11,-12,8,-20,12,-8,20,-6,20,-26,8,-8,15,-10,16,-18,12,

%U -16,20,-10,16,-16,8,-24,24,-8,21,-26,8,-20,20,-14,8,-28

%N Expansion of psi(-x)^2 * phi(x^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="/A246835/b246835.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^(-1/4) * eta(q)^2 * eta(q^4)^7 / (eta(q^2)^4 * eta(q^8)^2) in powers of q.

%F Euler transform of period 8 sequence [ -2, 2, -2, -5, -2, 2, -2, -3, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 16 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246836.

%F a(n) = (-1)^n * A213625(n). a(2*n) = A213622(n). a(2*n + 1) = -2 * A132969(n).

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

%e G.f. = q - 2*q^5 + 3*q^9 - 6*q^13 + 4*q^17 - 4*q^21 + 7*q^25 - 2*q^29 + ...

%t a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q^2]* EllipticTheta[2, 0, I*q^(1/2)]^2/(4*(-q)^(1/4)), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Nov 29 2017 *)

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^7 / (eta(x^2 + A)^4 * eta(x^8 + A)^2), n))};

%Y Cf. A132969, A213622, A213625, A246836.

%K sign

%O 0,2

%A _Michael Somos_, Sep 04 2014