cp's OEIS Frontend

A263456 Expansion of phi(-q) * f(-q^8)^3 / f(-q^24) in powers of q where phi(), f() are Ramanujan theta functions.

%I A263456 #10 Feb 16 2025 08:33:27
%S A263456 1,-2,0,0,2,0,0,0,-3,4,0,0,-6,0,0,0,2,6,0,0,0,0,0,0,0,-14,0,0,12,0,0,
%T A263456 0,-3,0,0,0,-4,0,0,0,12,6,0,0,-6,0,0,0,-6,-14,0,0,0,0,0,0,-6,24,0,0,0,
%U A263456 0,0,0,2,12,0,0,-6,0,0,0,-12,-24,0,0,12,0,0,0
%N A263456 Expansion of phi(-q) * f(-q^8)^3 / f(-q^24) in powers of q where phi(), f() are Ramanujan theta functions.
%C A263456 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A263456 G. C. Greubel, <a href="/A263456/b263456.txt">Table of n, a(n) for n = 0..2500</a>
%H A263456 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A263456 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A263456 Expansion of eta(q)^2 * eta(q^8)^3 / (eta(q^2) * eta(q^24)) in powers of q.
%F A263456 Euler transform of period 24 sequence [ -2, -1, -2, -1, -2, -1, -2, -4, -2, -1, -2, -1, -2, -1, -2, -4, -2, -1, -2, -1, -2, -1, -2, -3, ...].
%F A263456 G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 279936^(1/2) (t/I)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263452.
%F A263456 a(4*n + 2) = a(4*n + 3) = a(8*n + 5) = a(9*n + 6) = 0.
%e A263456 G.f. = 1 - 2*x + 2*x^4 - 3*x^8 + 4*x^9 - 6*x^12 + 2*x^16 + 6*x^17 - 14*x^25 + ...
%t A263456 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] QPochhammer[ q^8]^3 / QPochhammer[ q^24], {q, 0, n}];
%o A263456 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^8 + A)^3 / (eta(x^2 + A) * eta(x^24 + A)), n))};
%Y A263456 Cf. A263452.
%K A263456 sign
%O A263456 0,2
%A A263456 _Michael Somos_, Oct 19 2015