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A226194 Expansion of f(-x^1, -x^7) * f(-x^3, -x^5) in powers of x where f(, ) is Ramanujan's general theta function.

%I A226194 #18 Feb 16 2025 08:33:19
%S A226194 1,-1,0,-1,1,-1,1,-1,0,0,2,0,1,-1,1,-2,0,0,1,-1,0,-1,1,0,1,-2,0,-2,1,
%T A226194 0,1,0,1,-1,1,0,1,0,0,-1,3,-1,0,-1,0,-2,1,0,1,-1,1,0,1,0,0,-2,0,-1,0,
%U A226194 -1,2,-2,0,-1,0,0,2,-1,1,-1,2,0,0,0,0,-1,1,0,2
%N A226194 Expansion of f(-x^1, -x^7) * f(-x^3, -x^5) in powers of x where f(, ) is Ramanujan's general theta function.
%C A226194 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A226194 G. C. Greubel, <a href="/A226194/b226194.txt">Table of n, a(n) for n = 0..2500</a>
%H A226194 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A226194 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A226194 Expansion of q^(-5/8) * eta(q) * eta(q^8)^2 / eta(q^2) in powers of q.
%F A226194 Euler transform of period 8 sequence [-1, 0, -1, 0, -1, 0, -1, -2, ...].
%F A226194 a(n) = -I/2 * b(8*n + 5) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = (-1)^(e * (p+1)/8) * (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = (e+1) * I^(e * (p-1)/4) if p == 1 (mod 4).
%F A226194 G.f.: Product_{k>0} (1 - x^(8*k))^2 / (1 + x^k).
%F A226194 a(9*n + 2) = a(9*n + 8) = 0. a(9*n + 5) = -a(n).
%F A226194 a(n) = (-1)^n * A053692(n).
%e A226194 G.f. = 1 - x - x^3 + x^4 - x^5 + x^6 - x^7 + 2*x^10 + x^12 - x^13 + x^14 - 2*x^15 + ...
%e A226194 G.f. = q^5 - q^13 - q^29 + q^37 - q^45 + q^53 - q^61 + 2*q^85 + q^101 - q^109 + ...
%t A226194 a[ n_] := SeriesCoefficient[ QPochhammer[ q^8]^2 / QPochhammer[ -q, q], {q, 0, n}];
%o A226194 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^8 + A)^2 / eta(x^2 + A), n))};
%o A226194 (PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 8*n + 5; A = factor(n); simplify( -I/2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p%4 == 3, if( e%2, 0, (-1)^(e * (p+1) / 8)), (e+1) * I^(e * (p-1) / 4)))))};
%Y A226194 Cf. A053692, A226192.
%K A226194 sign
%O A226194 0,11
%A A226194 _Michael Somos_, May 30 2013