This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A139032 #14 Feb 16 2025 08:33:08
%S A139032 1,1,0,0,-1,0,0,0,0,0,2,0,0,-2,0,0,-1,0,0,4,0,0,-4,0,0,-1,0,0,8,0,0,
%T A139032 -8,0,0,-2,0,0,14,0,0,-14,0,0,-4,0,0,24,0,0,-23,0,0,-6,0,0,40,0,0,-38,
%U A139032 0,0,-10,0,0,63,0,0,-60,0,0,-16,0,0,98,0,0,-92,0,0,-24,0,0,150,0,0,-140,0,0,-36,0,0,224,0,0,-208
%N A139032 Expansion of 1 + c(q^6) / c(q^3) where c() is a cubic AGM theta function.
%C A139032 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A139032 G. C. Greubel, <a href="/A139032/b139032.txt">Table of n, a(n) for n = 0..1000</a>
%H A139032 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A139032 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A139032 Expansion of 1 + eta(q^3) * eta(q^18)^3 / (eta(q^6) * eta(q^9)^3) in powers of q.
%F A139032 Expansion of 1 + q * chi(-q^3) / chi(-q^9)^3 = psi(q) * chi(-q^3) / phi(-q^9) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
%F A139032 Expansion of eta(q^2)^2 * eta(q^3) * eta(q^18) / (eta(q) * eta(q^6) * eta(q^9)^2) in powers of q.
%F A139032 Euler transform of period 18 sequence [ 1, -1, 0, -1, 1, -1, 1, -1, 2, -1, 1, -1, 1, -1, 0, -1, 1, 0, ...].
%F A139032 G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = (3/2) * g(t) where q = exp(2 Pi i t) and g() is g.f. for A145977.
%F A139032 G.f.: Product_{k>0} P(2, x^k)^2 * P(18, x^k) / (P(3, x^k) * P(9, x^k)) where P(n, x) is n-th cyclotomic polynomial.
%F A139032 a(3*n) = 0 unless n = 0. a(3*n + 2) = 0. a(3*n + 1) = A092848(n). Convolution inverse of A145977.
%e A139032 1 + q - q^4 + 2*q^10 - 2*q^13 - q^16 + 4*q^19 - 4*q^22 - q^25 + 8*q^28 + ...
%t A139032 eta[x_] := QPochhammer[x]; A139032[n_] := SeriesCoefficient[eta[q^2]^2 *eta[q^3]*eta[q^18]/(eta[q]*eta[q^6]*eta[q^9]^2), {q, 0, n}];
%t A139032 Table[A139032[n], {n, 0, 20}] (* _G. C. Greubel_, Aug 09 2017 *)
%o A139032 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^18 + A) / (eta(x + A) * eta(x^6 + A) * eta(x^9 + A)^2), n))}
%Y A139032 Cf. A092848, A145977.
%K A139032 sign
%O A139032 0,11
%A A139032 _Michael Somos_, Apr 07 2008