A234565 Expansion of f(-q^3)^2 * Q(q^3) + 48 * q * f(-q^3)^10 in powers of q.
1, 48, 0, 238, -480, 0, 1679, 1680, 0, 2162, -1440, 0, 2401, -5040, 0, -6958, 11424, 0, -1442, 0, 0, -23040, -12480, 0, 1918, -7920, 0, -9362, 6720, 0, 14641, 50592, 0, 0, -36960, 0, 80640, -28560, 0, -20398, 0, 0, 28083, -34320, 0, 64078, 103776, 0, -38398
Offset: 0
Keywords
Examples
G.f. = 1 + 48*x + 238*x^3 - 480*x^4 + 1679*x^6 + 1680*x^7 + 2162*x^9 + ... G.f. = q + 48*q^5 + 238*q^13 - 480*q^17 + 1679*q^25 + 1680*q^29 + 2162*q^37 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..2500
Programs
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Mathematica
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 *) -
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))} -
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 )))))}
Formula
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).
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).
Comments