A230277 - OEIS

%I #10 Mar 12 2021 22:24:47

%S 1,2,0,-4,8,0,0,8,0,16,0,0,-10,0,0,-16,-16,0,0,-32,0,0,0,0,39,-20,0,0,

%T -40,0,0,32,0,-32,0,0,-70,0,0,64,80,0,0,0,0,0,0,0,49,78,0,40,56,0,0,0,

%U 0,-80,0,0,-22,0,0,-64,-80,0,0,64,0,0,0,0,110

%N Expansion of f(-q^6)^6 * ( q * chi(-q^3)^4 + 2*q^2 / chi(-q^3)^4 ) in powers of q where chi(), f() are Ramanujan theta functions.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A230277/b230277.txt">Table of n, a(n) for n = 1..2500</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of (eta(q^3)^2 * eta(q^6))^2 + 2 * (eta(q^6)^5 / eta(q^3)^2)^2 in powers of q.

%F a(n) is multiplicative with a(2^e) = -(-2)^e if e>0,  a(3^e) = 0^e, a(p^e) = p^e * (1 + (-1)^e)/2 if p == 3, 7 (mod 8), a(p^e) = a(p)*a(p^(e-1)) - p^2*a(p^(e-2)) and a(p) = t * (-1)^(t mod 3) if p == 1, 5 (mod 8) where t = 2 * if( p == 5 (mod 6) then 4*x*y else p - 8*y^2 ) and p = x^2 + 4*y^2.

%F a(2*n) = -2 * a(n). a(3*n) = a(4*n + 3) = 0.

%F a(3*n + 1) = A230280(n). a(3*n + 2) = 2 * A230278(n).

%e G.f. = q + 2*q^2 - 4*q^4 + 8*q^5 + 8*q^8 + 16*q^10 - 10*q^13 - 16*q^16 + ...

%t a[ n_] := SeriesCoefficient[ With[{t = QPochhammer[ q^3, q^6]^4}, QPochhammer[ q^6]^6 (q t + 2 q^2 / t)], {q, 0, n}]

%o (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^3 + A)^2 * eta(x^6 + A))^2 + 2 * x * (eta(x^6 + A)^5 / eta(x^3 + A)^2)^2, n))}

%o (PARI) {a(n) = local(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor( n); prod( k=1, matsize(A)[1], if( p = A[k, 1], e = A[k, 2]; if( p==3, 0, if( p==2, -(-2)^e, if( p%8==3 || p%8==7, if(e%2, 0, p^e), for( i = 1, sqrtint( p\4), if( issquare( p - 4*i^2, &x), y = i; break)); y = 2 * if( p%6 == 5, 4*x*y, p - 8*y^2); a1 = y = if( y%3==2, y, -y); a0 = 1; for( i=2, e, x = y*a1 - p^2*a0; a0 = a1; a1 = x); a1))))))}

%Y Cf. A230278, A230280.

%K sign,mult

%O 1,2

%A _Michael Somos_, Oct 15 2013