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A279476 Expansion of phi(-x^4) / (chi(-x^12) * f(-x)^2) in powers of x where phi(), chi(), f() are Ramanujan theta functions.

%I A279476 #15 Feb 16 2025 08:33:38
%S A279476 1,2,5,10,18,32,55,90,145,228,351,532,796,1172,1708,2462,3512,4966,
%T A279476 6965,9688,13383,18362,25031,33922,45717,61280,81737,108506,143387,
%U A279476 188672,247249,322734,419702,543852,702300,903932,1159779,1483492,1892012,2406210,3051796
%N A279476 Expansion of phi(-x^4) / (chi(-x^12) * f(-x)^2) in powers of x where phi(), chi(), f() are Ramanujan theta functions.
%C A279476 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A279476 G. C. Greubel, <a href="/A279476/b279476.txt">Table of n, a(n) for n = 0..5000</a>
%H A279476 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A279476 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A279476 Expansion of chi(x^2) / (chi(-x^12) * phi(-x)) in powers of x where phi(), chi() are Ramanujan theta functions.
%F A279476 Expansion of q^(-5/12) * eta(q^4)^2 * eta(q^24) / (eta(q)^2 * eta(q^8) * eta(q^12)) in powers of q.
%F A279476 Euler transform of period 24 sequence [ 2, 2, 2, 0, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 0, 2, 2, 2, 1, ...].
%F A279476 G.f. is a period 1 Fourier series that satisfies f(-1 / (864 t)) = 288^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A279479.
%F A279476 a(n) = A131945(2*n + 1).
%F A279476 a(n) ~ sqrt(5) * exp(sqrt(10*n)*Pi/3) / (24*n). - _Vaclav Kotesovec_, Nov 29 2019
%e A279476 G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 18*x^4 + 32*x^5 + 55*x^6 + ...
%e A279476 G.f. = q^5 + 2*q^17 + 5*q^29 + 10*q^41 + 18*q^53 + 32*q^65 + ...
%t A279476 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^4] QPochhammer[ -x^12, x^12] / QPochhammer[ x]^2 , {x, 0, n}];
%t A279476 a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^4] QPochhammer[ -x^12, x^12] / EllipticTheta[ 4, 0, x] , {x, 0, n}];
%o A279476 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 * eta(x^24 + A) / (eta(x + A)^2 * eta(x^8 + A) * eta(x^12 + A)), n))};
%o A279476 (PARI) q='q+O('q^99); Vec(eta(q^4)^2*eta(q^24)/(eta(q)^2*eta(q^8)*eta(q^12))) \\ _Altug Alkan_, Jul 30 2018
%Y A279476 Cf. A131945, A279479.
%K A279476 nonn
%O A279476 0,2
%A A279476 _Michael Somos_, Dec 12 2016