A186115 - OEIS

%I #16 Mar 12 2021 22:24:46

%S 1,0,2,-3,4,-6,9,-12,16,-21,28,-36,47,-60,76,-96,120,-150,185,-228,

%T 280,-342,416,-504,608,-732,878,-1050,1252,-1488,1765,-2088,2464,

%U -2901,3408,-3996,4676,-5460,6364,-7404,8600,-9972,11545,-13344,15400,-17748,20424

%N Expansion of q^(-1) * f(-q^3) * phi(-q^3) / (phi(-q^2) * psi(-q^9)) in powers of q where f(), phi(), psi() are Ramanujan theta functions.

%C Denoted by "(36~q)" in Simon Norton's replicable function list.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A186115/b186115.txt">Table of n, a(n) for n = -1..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 eta(q^3)^3 * eta(q^4) * eta(q^18) / (eta(q^2)^2 * eta(q^6) * eta(q^9) * eta(q^36)) in powers of q.

%F Euler transform of period 36 sequence [ 0, 2, -3, 1, 0, 0, 0, 1, -2, 2, 0, -1, 0, 2, -3, 1, 0, 0, 0, 1, -3, 2, 0, -1, 0, 2, -2, 1, 0, 0, 0, 1, -3, 2, 0, 0, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A261154. - _Michael Somos_, Aug 10 2015

%F a(n) = -(-1)^n * A058647(n).

%F a(2*n) = -3 * A128129(n). a(3*n) = 4 * A228447(n). - _Michael Somos_, Aug 10 2015

%e G.f. = 1/q + 2*q - 3*q^2 + 4*q^3 - 6*q^4 + 9*q^5 - 12*q^6 + 16*q^7 + ...

%t a[ n_] := SeriesCoefficient[ 2 q^(1/8) QPochhammer[ q^3] EllipticTheta[ 4, 0, q^3] / (EllipticTheta[ 4, 0, q^2] EllipticTheta[ 2, 0, q^(9/2)]), {q, 0, n}]; (* _Michael Somos_, Aug 10 2015 *)

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^4 + A) * eta(x^18 + A) / (eta(x^2 + A)^2 * eta(x^6 + A) * eta(x^9 + A) * eta(x^36 + A)), n))};

%Y Cf. A058647, A228447, A261154.

%K sign

%O -1,3

%A _Michael Somos_, Feb 13 2011