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A246752 Expansion of phi(-x) * chi(x) * psi(-x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

%I A246752 #15 Feb 16 2025 08:33:23
%S A246752 1,-1,-2,0,2,3,-2,0,1,-2,-2,0,2,0,-2,0,3,-2,0,0,2,3,-2,0,2,-2,-2,0,0,
%T A246752 0,-4,0,2,-1,-2,0,2,6,0,0,1,-2,-2,0,4,0,-2,0,0,-2,-2,0,2,0,-2,0,3,-2,
%U A246752 -2,0,2,0,0,0,2,-3,-2,0,0,6,-2,0,4,0,-2,0,2,0
%N A246752 Expansion of phi(-x) * chi(x) * psi(-x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
%C A246752 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A246752 G. C. Greubel, <a href="/A246752/b246752.txt">Table of n, a(n) for n = 0..1000</a>
%H A246752 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A246752 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A246752 Expansion of phi(-x) * f(x^1, x^5) in powers of x where phi(), f() are Ramanujan theta functions.
%F A246752 Expansion of q^(-1/3) * eta(q) * eta(q^2) * eta(q^3) * eta(q^12) / (eta(q^4) * eta(q^6)) in powers of q.
%F A246752 G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 768^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246838.
%F A246752 a(n) = (-1)^n * A246650(n).
%F A246752 Convolution of A002448 and A089801.
%F A246752 a(2*n) = A129451(n). a(4*n) = A123884(n). a(4*n + 1) = - A122861(n). a(4*n + 2) = - 2 * A121361(n). a(4*n + 3) = 0.
%e A246752 G.f. = 1 - x - 2*x^2 + 2*x^4 + 3*x^5 - 2*x^6 + x^8 - 2*x^9 - 2*x^10 + ...
%e A246752 G.f. = q - q^4 - 2*q^7 + 2*q^13 + 3*q^16 - 2*q^19 + q^25 - 2*q^28 + ...
%t A246752 a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^2] QPochhammer[ x^3] QPochhammer[ x^12] / (QPochhammer[ x^4] QPochhammer[ x^6]), {x, 0, n}];
%t A246752 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ -x, x^6] QPochhammer[ -x^5, x^6] QPochhammer[ x^6], {x, 0, n}];
%o A246752 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A) / (eta(x^4 + A) * eta(x^6 + A)), n))};
%Y A246752 Cf. A002448, A089801, A121361, A122861, A123884, A129451, A246650, A246838.
%K A246752 sign
%O A246752 0,3
%A A246752 _Michael Somos_, Sep 02 2014