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A261884 Expansion of (a(q) - a(q^2) - 2*a(q^3) + 2*a(q^6)) / 6 in powers of q where a() is a cubic AGM function.

%I A261884 #19 Nov 23 2023 08:00:46
%S A261884 1,-1,-1,1,0,1,2,-1,-1,0,0,-1,2,-2,0,1,0,1,2,0,-2,0,0,1,1,-2,-1,2,0,0,
%T A261884 2,-1,0,0,0,-1,2,-2,-2,0,0,2,2,0,0,0,0,-1,3,-1,0,2,0,1,0,-2,-2,0,0,0,
%U A261884 2,-2,-2,1,0,0,2,0,0,0,0,1,2,-2,-1,2,0,2,2,0
%N A261884 Expansion of (a(q) - a(q^2) - 2*a(q^3) + 2*a(q^6)) / 6 in powers of q where a() is a cubic AGM function.
%C A261884 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A261884 G. C. Greubel, <a href="/A261884/b261884.txt">Table of n, a(n) for n = 1..1000</a>
%F A261884 Moebius transform is period 18 sequence [ 1, -2, -2, 2, -1, 4, 1, -2, 0, 2, -1, -4, 1, -2, 2, 2, -1, 0, ...].
%F A261884 a(n) is multiplicative with a(2^e) = (-1)^e, a(3^e) = -1 if e>0, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
%F A261884 G.f.: Sum_{k>0} F(x^(6*k - 5)) - F(x^(6*k - 3)) + F(x^(6*k - 1)) where F(x) := x / (1 + x + x^2).
%F A261884 a(n) = A093829(n) unless n == 0 (mod 3). a(2*n) = - a(n). a(3*n + 1) = A033687(n).
%F A261884 a(6*n + 1) = A097195(n). a(6*n + 3) = - A033762(n). a(6*n + 5) = 0.
%F A261884 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(18*sqrt(3)) = 0.100766631346... . - _Amiram Eldar_, Nov 23 2023
%e A261884 G.f. = x - x^2 - x^3 + x^4 + x^6 + 2*x^7 - x^8 - x^9 - x^12 + 2*x^13 + ...
%t A261884 A004016[q_] := (QPochhammer[q]^3 + 9*q*QPochhammer[q^9]^3)/ QPochhammer[q^3]; A261884[n_] := SeriesCoefficient[(A004016[q] - A004016[q^2] - 2*A004016[q^3] + 2*A004016[q^6])/6, {q, 0, n}]; Table[A261884[n], {n, 1, 50}] (* _G. C. Greubel_, Sep 24 2017 *)
%o A261884 (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)^e, p==3, -1, p%6==1, e+1, 1-e%2)))};
%o A261884 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^9 + A)^3 / eta(x^3 + A) - x * eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^18 + A)^4 / (eta(x + A) * eta(x^6 + A)^2 * eta(x^9 + A)^2), n))};
%Y A261884 Cf. A033687, A033762, A093829, A097195, A035178 (apparently gives the absolute values).
%Y A261884 Cf. A004016, A005882, A005928.
%K A261884 sign,easy,mult
%O A261884 1,7
%A A261884 _Michael Somos_, Sep 04 2015