A262150 - OEIS

%I #9 Mar 12 2021 22:24:48

%S 1,0,1,-3,4,-3,5,-12,14,-10,18,-37,41,-34,54,-98,109,-92,138,-237,260,

%T -230,329,-531,583,-526,728,-1129,1233,-1143,1537,-2292,2503,-2355,

%U 3097,-4486,4889,-4677,6031,-8502,9263,-8962,11372,-15680,17066,-16703,20893

%N Expansion of f(-x^3)^3 / (f(-x^2) * f(-x^4)^2) in powers of x where f() is a Ramanujan theta function.

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

%H Vaclav Kotesovec, <a href="/A262150/b262150.txt">Table of n, a(n) for n = 0..10000</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^(1/24) * eta(q^3)^3 / (eta(q^2) * eta(q^4)^2) in powers of q.

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

%F a(n) = A143066(3*n).

%F a(n) ~ (-1)^n * exp(Pi*sqrt(n/2)) / (2^(3/4) * 3^(3/2) * n^(3/4)). - _Vaclav Kotesovec_, Sep 13 2015

%e G.f. = 1 + x^2 - 3*x^3 + 4*x^4 - 3*x^5 + 5*x^6 - 12*x^7 + 14*x^8 + ...

%e G.f. = q^-1 + q^47 - 3*q^71 + 4*q^95 - 3*q^119 + 5*q^143 - 12*q^167 + ...

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

%t nmax = 60; CoefficientList[Series[Product[(1-x^(3*k))^3 / ((1-x^(2*k))^3 * (1+x^(2*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 13 2015 *)

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

%Y Cf. A143066.

%K sign

%O 0,4

%A _Michael Somos_, Sep 13 2015