cp's OEIS Frontend

A208457 Expansion of x * f(-x) * f(-x^12)^3 * psi(-x^3) / psi(x^2) in powers of x where psi(), f() are Ramanujan theta functions.

%I A208457 #12 Feb 16 2025 08:33:16
%S A208457 0,1,-1,-2,0,3,2,-4,0,5,-1,-8,0,7,4,-8,0,9,-8,-10,0,14,6,-12,0,16,-6,
%T A208457 -14,0,15,8,-20,0,17,-14,-18,0,19,10,-24,0,26,-1,-22,0,23,16,-28,0,25,
%U A208457 -20,-32,0,32,14,-28,0,29,-12,-30,0,38,16,-32,0,33,-31
%N A208457 Expansion of x * f(-x) * f(-x^12)^3 * psi(-x^3) / psi(x^2) in powers of x where psi(), f() are Ramanujan theta functions.
%C A208457 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A208457 G. C. Greubel, <a href="/A208457/b208457.txt">Table of n, a(n) for n = 0..2500</a>
%H A208457 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A208457 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A208457 Expansion of q^(-2/3) * eta(q) * eta(q^2) * eta(q^3) * eta(q^12)^4 / (eta(q^4)^2 * eta(q^6)) in powers of q.
%F A208457 Euler transform of period 12 sequence [-1, -2, -2, 0, -1, -2, -1, 0, -2, -2, -1, -4, ...].
%F A208457 a(4*n) = 0. a(n) = -(-1)^n * A208435(n).
%e A208457 G.f. = x - x^2 - 2*x^3 + 3*x^5 + 2*x^6 - 4*x^7 + 5*x^9 - x^10 - 8*x^11 + ...
%e A208457 G.f. = q^5 - q^8 - 2*q^11 + 3*q^17 + 2*q^20 - 4*q^23 + 5*q^29 - q^32 - 8*q^35 + ...
%t A208457 QP:= QPochhammer; Join[{0}, CoefficientList[Series[Simplify[QP[q]* QP[q^2]*QP[q^3]*QP[q^12]^4/(QP[q^4]^2*QP[q^6]), q > 0], {q, 0, 50}], q]] (* _G. C. Greubel_, Aug 12 2018 *)
%o A208457 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)^4 / (eta(x^4 + A)^2 * eta(x^6 + A)), n))};
%Y A208457 Cf. A208435.
%K A208457 sign
%O A208457 0,4
%A A208457 _Michael Somos_, Feb 26 2012