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A258741 Expansion of f(x^3, x^5) / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.

%I A258741 #25 Feb 16 2025 08:33:25
%S A258741 1,-1,1,-1,2,-2,2,-3,4,-5,5,-6,8,-9,10,-12,15,-17,19,-22,26,-30,33,
%T A258741 -38,45,-51,56,-64,74,-83,92,-104,119,-133,147,-165,187,-208,229,-256,
%U A258741 288,-319,351,-390,435,-481,528,-584,649,-715,783,-863,954,-1047,1145
%N A258741 Expansion of f(x^3, x^5) / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.
%C A258741 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A258741 Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 16th equation.
%H A258741 G. C. Greubel, <a href="/A258741/b258741.txt">Table of n, a(n) for n = 0..1000</a>
%H A258741 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A258741 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A258741 Expansion of f(x^4, x^12) / f(x, x^7) where f(, ) is Ramanujan's general theta function.
%F A258741 Euler transform of period 16 sequence [ -1, 1, 0, 1, 0, 0, -1, 0, -1, 0, 0, 1, 0, 1, -1, 0, ...].
%F A258741 G.f.: 1 / (Product_{k>=0} (1 + x^(8*k + 1)) * (1 - x^(8*k + 4)) * (1 + x^(8*k + 7))).
%F A258741 G.f.: (1 + x^4 + x^12 + x^24 + x^40 + ...) / (1 + x + x^7 + x^10 + x^22 + ...). [Ramanujan]
%F A258741 G.f.: 1 - x * (1 - x) / (1 - x^2) + x^4 * (1 - x) * (1 - x^3) / ((1 - x^2) * (1 - x^4)) - ... [Ramanujan]
%F A258741 a(n) = (-1)^n * A036016(n) = A029838(2*n) = A082303(2*n).
%F A258741 Convolution product of A106507 and A214264.
%e A258741 G.f. = 1 - x + x^2 - x^3 + 2*x^4 - 2*x^5 + 2*x^6 - 3*x^7 + 4*x^8 - 5*x^9 + ...
%e A258741 G.f. = 1/q - q^15 + q^31 - q^47 + 2*q^63 - 2*q^79 + 2*q^95 - 3*q^111 + ...
%t A258741 a[ n_] := SeriesCoefficient[ QPochhammer[ -x^4, x^4] / (QPochhammer[ -x, x^8] QPochhammer[ -x^7, x^8]), {x, 0, n}];
%t A258741 a[ n_] := SeriesCoefficient[ 1 / Product[ (1 + x^(8 k + 1)) (1 - x^(8 k + 4)) (1 + x^(8 k + 7)), {k, 0, Ceiling[ n/8]}], {x, 0, n}];
%t A258741 a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{1, -1, 0, -1, 0, 0, 1, 0, 1, 0, 0, -1, 0, -1, 1, 0}[[Mod[k, 16, 1]]], {k, n}], {x, 0, n}];
%o A258741 (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 0, 1, -1, 0, -1, 0, 0, 1, 0, 1, 0, 0, -1, 0, -1, 1][k%16 + 1]), n))};
%Y A258741 Cf. A029838, A036016, A082303, A106507, A214264.
%K A258741 sign
%O A258741 0,5
%A A258741 _Michael Somos_, Nov 06 2015