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A129576 Expansion of phi(x) * chi(x) * psi(-x^3) in powers of x where phi(), chi(), psi() are Ramanujan theta functions.

%I A129576 #32 Feb 16 2025 08:33:05
%S A129576 1,3,2,0,2,3,2,0,1,6,2,0,2,0,2,0,3,6,0,0,2,3,2,0,2,6,2,0,0,0,4,0,2,3,
%T A129576 2,0,2,6,0,0,1,6,2,0,4,0,2,0,0,6,2,0,2,0,2,0,3,6,2,0,2,0,0,0,2,9,2,0,
%U A129576 0,6,2,0,4,0,2,0,2,0,0,0,2,6,4,0,0,3,4
%N A129576 Expansion of phi(x) * chi(x) * psi(-x^3) in powers of x where phi(), chi(), psi() are Ramanujan theta functions.
%C A129576 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A129576 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882). - _Michael Somos_, Jun 28 2017
%H A129576 G. C. Greubel, <a href="/A129576/b129576.txt">Table of n, a(n) for n = 0..1000</a>
%H A129576 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A129576 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A129576 From _Michael Somos_, Jun 28 2017: (Start)
%F A129576 Expansion of q^(-1/3) * (2*c(q) + c(-q)) / 3 = q^(-1/3) * (c(q) + 2*c(q^4)) / 3 in powers of q where c() is a cubic AGM theta function.
%F A129576 Expansion of (a(q) - a(q^3) + 2*a(q^4) - 2*a(q^12)) / 6 in powers of q where a() is a cubic AGM theta function.
%F A129576 Expansion of q^(-1/3) * eta(q^2)^7 * eta(q^3) * eta(q^12) / (eta(q)^3 * eta(q^4)^3 * eta(q^6)) in powers of q. (End)
%F A129576 Euler transform of period 12 sequence [3, -4, 2, -1, 3, -4, 3, -1, 2, -4, 3, -2, ...].
%F A129576 a(n) = b(3*n + 1) where b() is multiplicative and b(2^e) = 3 * (1 + (-1)^e) / 2 if e>0, a(3^e) = 0^e, a(p^e) = e+1 if p == 1 (mod 3), a(p^e) = (1 + (-1)^e)/2 if p == 2 (mod 3).
%F A129576 a(n) = A096936(3*n + 1) = A112298(3*n + 1).
%F A129576 2 * a(n) = A033716(3*n + 1). - _Michael Somos_, Sep 03 2016
%F A129576 a(n) = (-1)^n * A122161(n). - _Michael Somos_, Jun 28 2017
%F A129576 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(3) = 1.813799... (A093602). - _Amiram Eldar_, Dec 29 2023
%e A129576 G.f. = 1 + 3*x + 2*x^2 + 2*x^4 + 3*x^5 + 2*x^6 + x^8 + 6*x^9 + 2*x^10 + ...
%e A129576 G.f. = q + 3*q^4 + 2*q^7 + 2*q^13 + 3*q^16 + 2*q^19 + q^25 + 6*q^28 + ...
%t A129576 a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/8) EllipticTheta[ 3, 0, x] QPochhammer[ -x, x^2] EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}]; (* _Michael Somos_, Nov 11 2015 *)
%t A129576 a[ n_] := Length @ FindInstance[ x^2 + 3 y^2 == 3 n + 1, {x, y}, Integers, 10^9] / 2; (* _Michael Somos_, Sep 03 2016 *)
%t A129576 a[ n_] := If[ n < 1, Boole[n == 0], Times @@ (Which[# == 3, Boole[#2 == 0], # == 2, 3 (1 + (-1)^#2)/2, Mod[#, 3] == 2, (1 + (-1)^#2)/2, True, #2 + 1] & @@@ FactorInteger[3 n + 1])]; (* _Michael Somos_, Jun 28 2017 *)
%o A129576 (PARI) {a(n) = if( n<0, 0, n = 3*n + 1; sumdiv(n, d, kronecker(-3, d) * [0, 1, -2, 1] [n/d%4 + 1] ))};
%o A129576 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A)), n))};
%o A129576 (PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 3*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 3*(1-e%2), p==3, 0, p%3==2, 1-e%2, e+1)))}; /* _Michael Somos_, Jun 28 2017 */
%Y A129576 Cf. A033716, A093602, A096936, A112298, A122861.
%Y A129576 Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.
%K A129576 nonn,easy
%O A129576 0,2
%A A129576 _Michael Somos_, Apr 23 2007