A262152 - OEIS

%I #15 Mar 12 2021 22:24:48

%S 1,-1,1,-2,5,-6,4,-8,18,-20,16,-27,52,-58,47,-74,133,-146,127,-187,

%T 312,-343,304,-431,687,-751,687,-941,1436,-1569,1459,-1948,2879,-3139,

%U 2975,-3885,5569,-6071,5826,-7472,10457,-11385,11067,-13972,19122,-20813,20423

%N Expansion of f(-x^6)^3 / (f(-x^4)^2 * psi(x)) in powers of x where phi(), f() 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="/A262152/b262152.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^(-7/24) * eta(q) * eta(q^6)^3 / (eta(q^2)^2 * eta(q^4)^2) in powers of q.

%F Euler transform of period 12 sequence [-1, 1, -1, 3, -1, -2, -1, 3, -1, 1, -1, 0, ...].

%e G.f. = 1 - x + x^2 - 2*x^3 + 5*x^4 - 6*x^5 + 4*x^6 - 8*x^7 + 18*x^8 + ...

%e G.f. = q^7 - q^31 + q^55 - 2*q^79 + 5*q^103 - 6*q^127 + 4*q^151 - 8*q^175 + ...

%t a[ n_] := SeriesCoefficient[ 2 x^(1/8) QPochhammer[ x^6]^3 / (QPochhammer[ x^4]^2 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^6 + A)^3 / (eta(x^2 + A)^2 * eta(x^4 + A)^2), n))};

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

%K sign

%O 0,4

%A _Michael Somos_, Sep 13 2015