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A234565 Expansion of f(-q^3)^2 * Q(q^3) + 48 * q * f(-q^3)^10 in powers of q.

%I A234565 #21 Aug 08 2018 07:54:51
%S A234565 1,48,0,238,-480,0,1679,1680,0,2162,-1440,0,2401,-5040,0,-6958,11424,
%T A234565 0,-1442,0,0,-23040,-12480,0,1918,-7920,0,-9362,6720,0,14641,50592,0,
%U A234565 0,-36960,0,80640,-28560,0,-20398,0,0,28083,-34320,0,64078,103776,0,-38398
%N A234565 Expansion of f(-q^3)^2 * Q(q^3) + 48 * q * f(-q^3)^10 in powers of q.
%C A234565 f(-q) (g.f. A010815) and Q(q) (g.f. A004009) are Ramanujan q-series.
%H A234565 G. C. Greubel, <a href="/A234565/b234565.txt">Table of n, a(n) for n = 0..2500</a>
%F A234565 G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 12^5 (t/i)^5 f(t) where q = exp(2 Pi i t).
%F A234565 a(n) = b(4*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * p^(2*e) if p == 7 or 11 (mod 12), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) if p == 1 or 5 (mod 12).
%F A234565 a(3*n + 2) = 0. a(3*n) = A122266(n). a(3*n + 1) = 48 * A010818(n).
%e A234565 G.f. = 1 + 48*x + 238*x^3 - 480*x^4 + 1679*x^6 + 1680*x^7 + 2162*x^9 + ...
%e A234565 G.f. = q + 48*q^5 + 238*q^13 - 480*q^17 + 1679*q^25 + 1680*q^29 + 2162*q^37 + ...
%t A234565 eta[q_]:= q^(1/24)*QPochhammer[q]; Q:= (eta[q^3]^24 + 256*eta[q^6]^24)/( eta[q^3]*eta[q^6])^8; a:= CoefficientList[Series[q^(-1/4)*eta[q^3]^2*(48*q^(0/4)*eta[q^3]^8 + Q), {q, 0, 55}], q]]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Aug 07 2018 *)
%o A234565 (PARI) {a(n) = local(A, B); if( n<0, 0, A = x * O(x^n); B = 64 * x^3 * (eta(x^12 + A) / eta(x^3 + A))^8; polcoeff( 48 * x * eta(x^3 + A)^10 + (1 + 4*B + B^2) * eta(x^3 + A)^18 / eta(x^6 + A)^8, n))}
%o A234565 (PARI) {a(n) = local(A, p, e, i, x, y, a0, a1); if( n<0, 0, n = 4*n + 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k,1], e=A[k,2]; if( p<5, 0, if( p%12 > 6, if( e%2, 0, p^(2*e)), forstep( i = 1, sqrtint( p), 2, if( issquare( p - i^2, &y), x=i; break)); if( p%12 == 5, a1 = 8 * x*y * (x-y) * (x+y) * (-1)^((x%6==1) + (y%6==4)), a1 = 2 * (x^2-y^2+2*x*y) * (x^2-y^2-2*x*y) * (-1)^(x%6==3) ); a0 = 1; y = a1; for( i=2, e, x = y * a1 - p^4 * a0; a0=a1; a1=x); a1 )))))}
%Y A234565 Cf. A010818, A122266.
%K A234565 sign
%O A234565 0,2
%A A234565 _Michael Somos_, Jan 06 2014