%I #24 Feb 16 2025 08:33:26
%S 1,0,0,-1,1,-1,1,-1,2,-2,2,-3,4,-4,4,-6,7,-7,8,-10,12,-13,14,-17,21,
%T -22,24,-29,33,-36,40,-46,53,-58,63,-73,83,-90,99,-113,127,-138,152,
%U -171,191,-209,228,-255,285,-309,338,-377,416,-453,495,-547,603,-656
%N Expansion of f(x, x^7) / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 15th equation.
%H G. C. Greubel, <a href="/A259774/b259774.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="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of f(x^4, x^12) / f(x^3, x^5) where f(, ) is Ramanujan's general theta function.
%F Euler transform of period 16 sequence [ 0, 0, -1, 1, -1, 1, 0, 0, 0, 1, -1, 1, -1, 0, 0, 0, ...].
%F G.f.: (1 + x^4 + x^12 + x^24 + x^40 + ...) / (1 + x^3 + x^5 + x^14 + x^18 + ...). [Ramanujan]
%F G.f.: 1 - x^3 * (1 - x) / (1 - x^2) + x^8 * (1 - x) * (1 - x^3) / ((1 - x^2) * (1 - x^4)) - ... [Ramanujan]
%F a(n) = (-1)^n * A036015(n) = A029838(2*n + 1) = - A082303(2*n + 1).
%F Convolution product of A106507 and A214263.
%e G.f. = 1 - x^3 + x^4 - x^5 + x^6 - x^7 + 2*x^8 - 2*x^9 + 2*x^10 - 3*x^11 + ...
%e G.f. = q^7 - q^55 + q^71 - q^87 + q^103 - q^119 + 2*q^135 - 2*q^151 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x^4, x^4] / (QPochhammer[ -x^3, x^8] QPochhammer[ -x^5, x^8]), {x, 0, n}];
%t a[ n_] := SeriesCoefficient[ 1 / Product[ (1 + x^(8 k + 3)) (1 - x^(8 k + 4)) (1 + x^(8 k + 5)), {k, 0, Ceiling[ n/8]}], {x, 0, n}];
%t a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ 0, 0, 1, -1, 1, -1, 0, 0, 0, -1, 1, -1, 1, 0, 0, 0}[[Mod[k, 16, 1]]], {k, n}], {x, 0, n}];
%o (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 0, 0, 0, 1, -1, 1, -1, 0, 0, 0, -1, 1, -1, 1, 0, 0][k%16 + 1]), n))};
%Y Cf. A029838, A036015, A082303, A106507, A214263.
%K sign,changed
%O 0,9
%A _Michael Somos_, Nov 08 2015