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%I #14 Jan 22 2024 01:18:53
%S 1,1,-5,1,6,-5,8,1,-23,6,12,-5,14,8,-30,1,18,-23,20,6,-40,12,24,-5,31,
%T 14,-77,8,30,-30,32,1,-60,18,48,-23,38,20,-70,6,42,-40,44,12,-138,24,
%U 48,-5,57,31,-90,14,54,-77,72,8,-100,30,60,-30,62,32,-184
%N Expansion of (1 - b(q)*b(q^2)) / 3 where b() is a cubic AGM function. Expansion of (1 - eta(q)^3 * eta(q^2)^3 / (eta(q^3) * eta(q^6))) / 3 in powers of q.
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%D Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 84, Eq. (32.65).
%F Expansion of (1 - eta(q)^3 * eta(q^2)^3 / (eta(q^3) * eta(q^6))) / 3 in powers of q.
%F a(n) is multiplicative with a(2^e) = 1, a(3^e) = 4- 3^(e+1), a(p^e) = (p^(e+1) - 1) / (p-1) if p>3.
%F G.f.: (1 - Product_{k>0} ((1 - x^k) * (1 -x^(2*k)))^3 / ((1 - x^(3*k)) * (1 - x^(6*k)))) / 3.
%F G.f.: Sum_{k>0} (6*k-1) * x^(6*k-1) / (1 - x^(6*k-1)) - 2 * (6*k-5) * x^(6*k-3) / (1 - x^(6*k-3)) + (6*k-5) * x^(6*k-5) / (1 - x^(6*k-5)).
%F -3 * a(n) = A131943(n) unless n=0.
%F Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = 4*Pi^2/81 = 0.487387... . - _Amiram Eldar_, Jan 22 2024
%e G.f. = q + q^2 - 5*q^3 + q^4 + 6*q^5 - 5*q^6 + 8*q^7 + q^8 - 23*q^9 + 6*q^10 +...
%t a[ n_] := If[ n < 1, 0, Sum[ d {0, 1, 0, -2, 0, 1}[[ Mod[ d, 6] + 1]], {d, Divisors @ n}]]; (* _Michael Somos_, Nov 11 2015 *)
%t QP = QPochhammer; s = (1 - QP[q]^3*(QP[q^2]^3/(QP[q^3]*QP[q^6])))/(3 q) + O[q]^70; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 27 2015 *)
%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d*((d%6==1) + (d%6==5) - 2*(d%6==3))))};
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^3 * eta(x^2 + A)^3 / (eta(x^3 + A) * eta(x^6 + A)))/3, n))};
%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, 4 - p^(e+1), (p^(e+1) - 1) / (p-1) )))};
%Y Cf. A004016, A005882, A005928, A131943.
%K sign,mult
%O 1,3
%A _Michael Somos_, Jul 30 2007
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