A224833 - OEIS

%I #17 Mar 12 2021 22:24:46

%S 1,-5,8,-4,4,-13,12,-4,5,-16,24,-8,4,-20,12,-8,9,-20,32,-4,12,-29,12,

%T -8,8,-36,40,-8,8,-20,24,-16,8,-25,40,-12,12,-32,24,-12,13,-48,40,-8,

%U 8,-40,36,-8,16,-20,56,-16,12,-52,12,-20,13,-36,56,-16,20,-40,24

%N Expansion of phi(-x)^2 * chi(-x) * psi(x^3) in powers of x where phi(), psi(), chi() 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="/A224833/b224833.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^(-1/3) * eta(q)^5 * eta(q^6)^2 / (eta(q^2)^3 * eta(q^3)) in powers of q.

%F Euler transform of period 6 sequence [ -5, -2, -4, -2, -5, -3, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 73728^(1/2) (t / i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227595.

%F -2 * a(n) = A224822(3*n + 1).

%e 1 - 5*x + 8*x^2 - 4*x^3 + 4*x^4 - 13*x^5 + 12*x^6 - 4*x^7 + 5*x^8 - 16*x^9 + ...

%e q - 5*q^4 + 8*q^7 - 4*q^10 + 4*q^13 - 13*q^16 + 12*q^19 - 4*q^22 + 5*q^25 + ...

%t eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[q^(-1/3)* eta[q]^5*eta[q^6]^2/(eta[q^2]^3*eta[q^3]), {q, 0, n}];  Table[a[n], {n,0,50}] (* _G. C. Greubel_, Mar 19 2018 *)

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^5 * eta(x^6 + A)^2 / (eta(x^2 + A)^3 * eta(x^3 + A)), n))}

%Y Cf. A224822, A227595.

%K sign

%O 0,2

%A _Michael Somos_, Jul 21 2013