cp's OEIS Frontend

A260516 Expansion of f(x, x^2) * f(x^2, x^10) in powers of x where f(,) is Ramanujan's general theta function.

%I A260516 #11 Feb 16 2025 08:33:26
%S A260516 1,1,2,1,1,1,0,2,0,1,1,1,2,0,1,2,1,3,1,0,0,1,2,1,1,1,1,0,2,0,0,1,2,1,
%T A260516 1,1,1,2,1,1,1,0,3,1,2,1,0,2,0,1,1,2,0,1,2,0,1,2,1,1,0,1,0,0,1,0,1,4,
%U A260516 2,0,1,1,2,2,0,0,0,2,1,1,2,2,2,1,0,1,1
%N A260516 Expansion of f(x, x^2) * f(x^2, x^10) in powers of x where f(,) is Ramanujan's general theta function.
%C A260516 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A260516 G. C. Greubel, <a href="/A260516/b260516.txt">Table of n, a(n) for n = 0..2500</a>
%H A260516 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A260516 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A260516 Expansion of q^(-17/24) * eta(q^3)^2 * eta(q^4)^2 * eta(q^24) / (eta(q) * eta(q^8) * eta(q^12)) in powers of q.
%F A260516 Euler transform of period 24 sequence [ 1, 1, -1, -1, 1, -1, 1, 0, -1, 1, 1, -2, 1, 1, -1, 0, 1, -1, 1, -1, -1, 1, 1, -2, ...].
%e A260516 G.f. = 1 + x + 2*x^2 + x^3 + x^4 + x^5 + 2*x^7 + x^9 + x^10 + x^11 + ...
%e A260516 G.f. = q^17 + q^41 + 2*q^65 + q^89 + q^113 + q^137 + 2*q^185 + q^233 + ...
%t A260516 a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^2 QPochhammer[ -x^2, x^4] / (QPochhammer[ x, x^2] QPochhammer[ x^12, x^24]), {x, 0, n}];
%t A260516 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] EllipticTheta[ 4, 0, x^4] EllipticTheta[ 2, Pi/4, x^3] / (x^(3/4) Sqrt[2] QPochhammer[ x]), {x, 0, n}];
%o A260516 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^4 + A)^2 * eta(x^24 + A) / (eta(x + A) * eta(x^8 + A) * eta(x^12 + A)), n))};
%o A260516 (PARI) q='q+O('q^99); Vec(eta(q^3)^2*eta(q^4)^2*eta(q^24)/(eta(q)*eta(q^8)*eta(q^12))) \\ _Altug Alkan_, Aug 01 2018
%K A260516 nonn
%O A260516 0,3
%A A260516 _Michael Somos_, Jul 27 2015