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A182057 Expansion of psi(x) * f(x^4) / (psi(x^3) * f(x^6) * chi(-x^24)) in powers of x where psi(), chi(), f() are Ramanujan theta functions.

%I A182057 #16 Feb 16 2025 08:33:13
%S A182057 1,1,0,0,0,1,0,0,-2,-3,0,0,4,1,0,0,-1,5,0,0,-8,-10,0,0,14,4,0,0,-4,17,
%T A182057 0,0,-23,-31,0,0,40,9,0,0,-10,46,0,0,-60,-79,0,0,98,21,0,0,-24,112,0,
%U A182057 0,-140,-183,0,0,224,46,0,0,-54,249,0,0,-304,-396,0
%N A182057 Expansion of psi(x) * f(x^4) / (psi(x^3) * f(x^6) * chi(-x^24)) in powers of x where psi(), chi(), f() are Ramanujan theta functions.
%C A182057 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A182057 G. C. Greubel, <a href="/A182057/b182057.txt">Table of n, a(n) for n = 0..1000</a>
%H A182057 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A182057 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A182057 Expansion of q^(-2/3) * eta(q^2)^2 * eta(q^3) * eta(q^8)^3 * eta(q^48) / (eta(q) * eta(q^4) * eta(q^6) * eta(q^12)^3 * eta(q^16)) in powers of q.
%F A182057 Euler transform of period 48 sequence [1, -1, 0, 0, 1, -1, 1, -3, 0, -1, 1, 3, 1, -1, 0, -2, 1, -1, 1, 0, 0, -1, 1, 0, 1, -1, 0, 0, 1, -1, 1, -2, 0, -1, 1, 3, 1, -1, 0, -3, 1, -1, 1, 0, 0, -1, 1, 0, ...].
%F A182057 a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A182032(12*n + 2). a(4*n + 1) = A182032(12*n + 5).
%F A182057 Expansion of psi(x) * f(x^4) / (f(x^3) * phi(x^6) * chi(x^12)) in powers of x where phi(), psi(), f() are Ramanujan theta functions. - _Michael Somos_, Aug 10 2017
%e A182057 G.f. = 1 + x + x^5 - 2*x^8 - 3*x^9 + 4*x^12 + x^13 - x^16 + 5*x^17 - 8*x^20 + ...
%e A182057 G.f. = q^2 + q^5 + q^17 - 2*q^26 - 3*q^29 + 4*q^38 + q^41 - q^50 + 5*q^53 + ...
%t A182057 eta[x_] := x^(1/24)*QPochhammer[x]; A182057[n_] := SeriesCoefficient[ q^(-2/3)*eta[q^2]^2*eta[q^3]*eta[q^8]^3*eta[q^48]/(eta[q]*eta[q^4]* eta[q^6]*eta[q^12]^3 *eta[q^16]), {q, 0, n}]; Table[A182057[n], {n,0,50}] (* _G. C. Greubel_, Aug 10 2017 *)
%o A182057 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^8 + A)^3 * eta(x^48 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^12 + A)^3 * eta(x^16 + A)), n))}
%Y A182057 Cf. A182032.
%K A182057 sign
%O A182057 0,9
%A A182057 _Michael Somos_, Apr 08 2012