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A255648 Expansion of (a(q) + a(q^2) + a(q^3) + a(q^6) - 4) / 6 in powers of q where a() is a cubic AGM theta function.

%I A255648 #22 Feb 16 2025 08:33:25
%S A255648 1,1,2,1,0,2,2,1,2,0,0,2,2,2,0,1,0,2,2,0,4,0,0,2,1,2,2,2,0,0,2,1,0,0,
%T A255648 0,2,2,2,4,0,0,4,2,0,0,0,0,2,3,1,0,2,0,2,0,2,4,0,0,0,2,2,4,1,0,0,2,0,
%U A255648 0,0,0,2,2,2,2,2,0,4,2,0,2,0,0,4,0,2,0
%N A255648 Expansion of (a(q) + a(q^2) + a(q^3) + a(q^6) - 4) / 6 in powers of q where a() is a cubic AGM theta function.
%C A255648 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C A255648 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A255648 G. C. Greubel, <a href="/A255648/b255648.txt">Table of n, a(n) for n = 1..10000</a>
%H A255648 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A255648 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A255648 Expansion of (b(q^2)^2 / b(q) + b(q^6)^2 / b(q^3) - 2) / 3 in powers of q where b() is a cubic AGM theta function.
%F A255648 Expansion of (psi(q)^3 / psi(q^3) + psi(q^3)^3 / psi(q^9) - 2) / 3 in powers of q where psi() is a Ramanujan theta function.
%F A255648 Moebius transform is period 18 sequence [ 1, 0, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -1, 0, -1, 0, ...].
%F A255648 a(n) is multiplicative with a(2^e) = 1, a(3^e) = 2 if e>1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
%F A255648 G.f.: Sum_{k>0} (x^k + x^(3*k)) / (1 + x^(2*k))^2 + (x^(3*k) + x^(9*k)) / (1 + x^(6*k))^2.
%F A255648 a(2*n) = a(n). a(3*n) = 2 * A035178(n). a(3*n + 1) = A033687(n). a(6*n + 5) = 0.
%F A255648 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - _Amiram Eldar_, Dec 22 2023
%e A255648 G.f. = q + q^2 + 2*q^3 + q^4 + 2*q^6 + 2*q^7 + q^8 + 2*q^9 + 2*q^12 + ...
%t A255648 a[ n_] := If[ n < 1, 0, Sum[ { 1, 0, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -1, 0, -1, 0}[[Mod[ d, 18, 1]]], { d, Divisors[ n]}]];
%o A255648 (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, [ 0, 1, 0, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -1, 0, -1][d%18 + 1]))};
%o A255648 (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, 2, p%6==1, e+1, 1-e%2)))};
%Y A255648 Cf. A033687, A035178, A248897.
%Y A255648 Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.
%K A255648 nonn,easy,mult
%O A255648 1,3
%A A255648 _Michael Somos_, May 06 2015