cp's OEIS Frontend

A257657 Expansion of f(-x, -x) * f(-x^6, -x^6) / f(x, x^2) in powers of x where f(,) is Ramanujan's general theta function.

%I A257657 #12 Feb 16 2025 08:33:25
%S A257657 1,-3,2,1,-1,-1,3,-1,0,-2,-2,2,1,-3,3,4,-1,-3,1,0,-1,-2,0,3,1,-6,2,4,
%T A257657 -4,-1,4,2,-1,-3,0,5,-1,-9,5,7,-4,-7,4,5,-3,-4,0,8,-1,-13,4,11,-7,-7,
%U A257657 7,6,-1,-10,0,14,-1,-15,8,15,-10,-14,8,11,-7,-13,2,17
%N A257657 Expansion of f(-x, -x) * f(-x^6, -x^6) / f(x, x^2) in powers of x where f(,) is Ramanujan's general theta function.
%C A257657 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A257657 G. C. Greubel, <a href="/A257657/b257657.txt">Table of n, a(n) for n = 0..2500</a>
%H A257657 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A257657 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A257657 Expansion of q^(1/24) * eta(q)^3 * eta(q^6)^3 / (eta(q^2)^2 * eta(q^3)^2 * eta(q^12)) in powers of q.
%F A257657 Euler transform of period 12 sequence [ -3, -1, -1, -1, -3, -2, -3, -1, -1, -1, -3, -1, ...].
%F A257657 a(n) = 2 * A260413(n) - A053250(n).
%e A257657 G.f. = 1 - 3*x + 2*x^2 + x^3 - x^4 - x^5 + 3*x^6 - x^7 - 2*x^9 - 2*x^10 + ...
%e A257657 G.f. = 1/q - 3*q^23 + 2*q^47 + q^71 - q^95 - q^119 + 3*q^143 - q^167 + ...
%t A257657 a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^2 QPochhammer[ x] QPochhammer[ -x^3] / QPochhammer[ x^3], {x, 0, n}];
%o A257657 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^6 + A)^3 / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^12 + A)), n))};
%Y A257657 Cf. A053250, A260413.
%K A257657 sign
%O A257657 0,2
%A A257657 _Michael Somos_, Jul 26 2015