|
%I #16 Aug 26 2016 22:43:35
%S 1,24,20,0,-70,-192,56,0,-125,480,308,0,110,0,-520,0,57,-1680,0,0,182,
%T 1536,-880,0,1190,1344,884,0,0,0,-1400,0,-1330,-3000,1820,0,-646,
%U -3840,0,0,-1331,7392,380,0,1120,0,2576,0,0,2640,1748,0,-3850,0,-3400,0,2703,-12480,-2500,0
%N Expansion of q^(-1/3) * (eta(q)^8 + 32 * eta(q^4)^8) in powers of q.
%H G. C. Greubel, <a href="/A153729/b153729.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2592 (t / i)^4 g(t) where q = exp(2 Pi i t) and g() is g.f. for A153728.
%F a(4*n + 3) = 0.
%F a(n) = b(3*n + 1) where b(n) is multiplicative and b(3^e) = 0^e, b(2^e) = (-3/2) * (1+(-1)^e) * (-8)^(e/2), b(p^e) = (1/2) * (1+(-1)^e) * (-p^3) ^ (e/2) if p == 2 (mod 3), b(p^e) = b(p) * b(p^(e-1)) - p^3 * b(p^(e-2)) if p == 1 (mod 3) where b(p) = x * (x^2 -3*p), 4*p = x^2 + 3*y^2, |x|<|y| and x == 2 (mod 3). - _Michael Somos_, Mar 01 2011
%e q + 24*q^4 + 20*q^7 - 70*q^13 - 192*q^16 + 56*q^19 - 125*q^25 + ...
%t QP = QPochhammer; s=QP[q]^8+32*q*QP[q^4]^8 + O[q]^60; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 25 2015, adapted from PARI *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^8 + 32 * x * eta(x^4 + A)^8, n))}
%o (PARI) {a(n) = local(A, p, e, x, y, a0, a1); if( n<0, 0, n = 3*n + 1; 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, -3 * ((e+1)%2) * (-8)^(e\2), if( p%3==2, if(e%2, 0, (-p^3) ^ (e/2)), forstep( y = sqrtint(4*p\3), sqrtint(p\3), -1, if( issquare( 4*p - 3*y^2, &x), if( x%3!=2, x = -x); break)); a0 = 1; a1 = y = x * (x^2 - 3*p); for( i=2, e, x = y*a1 - p^3*a0; a0 = a1; a1 = x); a1))))))} /* _Michael Somos_, Mar 01 2011 */
%Y A000731(2*n) = A153728(n) = a(2*n). 24 * A000731(n) = a(4*n + 1).
%K sign
%O 0,2
%A _Michael Somos_, Dec 31 2008
|
