|
%I #14 Sep 07 2018 01:25:25
%S 1,-1,-1,-1,2,1,0,-1,1,-2,0,1,2,0,-2,-1,2,-1,0,-2,0,0,0,1,3,-2,-1,0,2,
%T 2,0,-1,0,-2,0,-1,2,0,-2,-2,2,0,0,0,2,0,0,1,1,-3,-2,-2,2,1,0,0,0,-2,0,
%U 2,2,0,0,-1,4,0,0,-2,0,0,0,-1,2,-2,-3,0,0,2,0,-2,1,-2,0,0,4,0,-2,0,2,-2,0,0,0,0,0,1,2,-1,0,-3,2,2,0,-2,0
%N Expansion of (theta_4(q^3)^2 - theta_4(q)^2)/4 in powers of q.
%H G. C. Greubel, <a href="/A121450/b121450.txt">Table of n, a(n) for n = 1..5000</a>
%F Expansion of eta(q) * eta(q^3) * eta(q^12)^3 /(eta(q^4) * eta(q^6)^2) in powers of q.
%F Euler transform of period 12 sequence [-1, -1, -2, 0, -1, 0, -1, 0, -2, -1, -1, -2, ...].
%F Moebius transform is period 24 sequence [1, -2, -2, 0, 1, 4, -1, 0, 2, -2, -1, 0, 1, 2, -2, 0, 1, -4, -1, 0, 2, 2, -1, 0, ...].
%F Multiplicative with a(2^e) = -1 if e > 0, a(3^e) = (-1)^e, a(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4), a(p^e) = e+1 if p == 1 (mod 4).
%F G.f.: Sum_{k>0} Kronecker(-4, k) * x^k * (1 - x^k)/ (1 + x^(3*k)) = x * Product_{k>0} (1 - x^k)^2 * (1 - x^k + x^(2*k)) * (1 + x^(2*k))^2 *(1 + x^k + x^(2*k))^2 * (1 - x^(2*k) + x^(4*k))^3.
%e G.f. = x - x^2 - x^3 - x^4 + 2*x^5 + x^6 - x^8 + x^9 - 2*x^10 + x^12 + ... - _Michael Somos_, Sep 07 2018
%t eta[q_]:= q^(1/24)*QPochhammer[q]; Rest[CoefficientList[Series[eta[q]* eta[q^3]*eta[q^12]^3/(eta[q^4]*eta[q^6]^2), {q, 0, 100}], q]] (* _G. C. Greubel_, Apr 19 2018 *)
%o (PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -1, p==3, (-1)^e, p%4==1, (e+1), (1+(-1)^e)/2)))};
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^12 + A)^3 / (eta(x^4 + A) * eta(x^6 + A)^2), n))};
%Y A035154(n) = |a(n)|.
%Y Apart from signs, same as A035154 and A113446.
%K sign,mult
%O 1,5
%A _Michael Somos_, Jul 30 2006
|
