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A260118 Expansion of f(-x, -x^5) * psi(x^4) in powers of x where psi(), f() are Ramanujan theta functions.

%I A260118 #10 Feb 16 2025 08:33:26
%S A260118 1,-1,0,0,1,-2,0,0,1,-1,0,0,2,-1,0,0,1,-1,0,0,2,-1,0,0,1,-2,0,0,1,-1,
%T A260118 0,0,1,-2,0,0,0,-1,0,0,3,-1,0,0,1,-3,0,0,1,0,0,0,1,0,0,0,2,-1,0,0,2,
%U A260118 -2,0,0,1,-2,0,0,2,-1,0,0,0,-1,0,0,1,-1,0,0,2
%N A260118 Expansion of f(-x, -x^5) * psi(x^4) in powers of x where psi(), f() are Ramanujan theta functions.
%C A260118 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A260118 G. C. Greubel, <a href="/A260118/b260118.txt">Table of n, a(n) for n = 0..1000</a>
%H A260118 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A260118 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A260118 Expansion of psi(-x^2)^2 * psi(x^3) / f(x) in powers of x where psi(), f() are Ramanujan theta functions.
%F A260118 Expansion of psi(-x) * psi(x^3) / chi(-x^4)^2 in powers of x where psi(), chi() are Ramanujan theta functions.
%F A260118 Expansion of q^(-5/6) * eta(q) * eta(q^6)^2 * eta(q^8)^2 / (eta(q^2) * eta(q^3) * eta(q^4)) in powers of q.
%F A260118 Euler transform of period 24 sequence [ -1, 0, 0, 1, -1, -1, -1, -1, 0, 0, -1, 0, -1, 0, 0, -1, -1, -1, -1, 1, 0, 0, -1, -2, ...].
%F A260118 a(n) = (-1)^n * A128582(n). 2 * a(n) = - A134177(3*n + 2) = A190615(3*n + 2).
%F A260118 a(4*n) = A128583(n). a(4*n + 1) = - A128591(n). a(4*n + 2) = a(4*n + 3) = 0.
%e A260118 G.f. = 1 - x + x^4 - 2*x^5 + x^8 - x^9 + 2*x^12 - x^13 + x^16 - x^17 + ...
%e A260118 G.f. = q^5 - q^11 + q^29 - 2*q^35 + q^53 - q^59 + 2*q^77 - q^83 + q^101 + ...
%t A260118 a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x]^2 EllipticTheta[ 2, 0, x^(3/2)]/(4 x^(7/8) QPochhammer[ -x]), {x, 0, n}];
%t A260118 a[ n_] := If[ n < 0, 0, With[ {m = 6 n + 5}, (-1)^n DivisorSum[ m, {-1, 0, 0, 0, 1, 0}[[Mod[#, 6, 1]]] {1, 0, 0, 0, 0, 0, 1, 0}[[Mod[m/#, 8, 1]]] &]]];
%o A260118 (PARI) {a(n) = my(m); if( n<0, 0, m=6*n + 5; (-1)^n * sumdiv(m, d, ((d%6==5) - (d%6==1)) * ((m/d%8==1) + (m/d%8==7))))};
%o A260118 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^2 * eta(x^8 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^4 + A)), n))};
%Y A260118 Cf. A128538, A128582, A128591, A134177, A190615.
%K A260118 sign
%O A260118 0,6
%A A260118 _Michael Somos_, Jul 16 2015