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%I #15 Oct 15 2022 08:09:41
%S 1,1,2,1,2,2,0,1,-2,2,0,2,2,0,4,1,2,-2,0,2,0,0,0,2,3,2,2,0,2,4,0,1,0,
%T 2,0,-2,2,0,4,2,2,0,0,0,-4,0,0,2,1,3,4,2,2,2,0,0,0,2,0,4,2,0,0,1,4,0,
%U 0,2,0,0,0,-2,2,2,6,0,0,4,0,2,-2,2,0,0,4,0,4,0,2,-4,0,0,0,0,0,2,2,1,0,3,2,4,0,2,0
%N Expansion of eta(q^3)^2 * eta(q^4) * eta(q^6)^2 * eta(q^36) / (eta(q) * eta(q^9) * eta(q^12)^2) in powers of q.
%H G. C. Greubel, <a href="/A122864/b122864.txt">Table of n, a(n) for n = 1..1000</a>
%F Euler transform of period 36 sequence [ 1, 1, -1, 0, 1, -3, 1, 0, 0, 1, 1, -2, 1, 1, -1, 0, 1, -2, 1, 0, -1, 1, 1, -2, 1, 1, 0, 0, 1, -3, 1, 0, -1, 1, 1, -2, ...].
%F Moebius transform is period 36 sequence [ 1, 0, 1, 0, 1, 0, -1, 0, -4, 0, -1, 0, 1, 0, 1, 0, 1, 0, -1, 0, -1, 0, -1, 0, 1, 0, 4, 0, 1, 0, -1, 0, -1, 0, -1, 0, ...].
%F a(n) is multiplicative with a(2^e) = 1, a(3^e) = 2*(-1)^(e+1) if e>0, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4).
%F a(3*n) = 2 * A113446(n). a(3*n + 1) = A002564(3*n + 1) = A035154(3*n + 1) = A113446(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - _Amiram Eldar_, Oct 15 2022
%e q + q^2 + 2*q^3 + q^4 + 2*q^5 + 2*q^6 + q^8 - 2*q^9 + 2*q^10 + 2*q^12 + ...
%t eta[x_] := x^(1/24)*QPochhammer[x]; A122864[n_] := SeriesCoefficient[ (eta[q^3]^2*eta[q^4]*eta[q^6]^2*eta[q^36])/(eta[q]*eta[q^9]*eta[q^12]^2), {q, 0, n}]; Table[A122864[n], {n, 50}] (* _G. C. Greubel_, Sep 16 2017 *)
%o (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^6 + A)^2 * eta(x^36 + A) /(eta(x + A) * eta(x^9 + A) * eta(x^12 + A)^2), n))}
%o (PARI) {a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 1, if( p==3, -2*(-1)^e, if( p%4==1, e+1, 1-e%2))))))}
%Y Cf. A002564, A019670, A035154, A113446, A122856, A122865.
%K sign,mult
%O 1,3
%A _Michael Somos_, Sep 15 2006
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