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A113661 Expansion of (phi(x)^3/phi(x^3) - 1)/6 where phi() is a Ramanujan theta function.

%I A113661 #34 Feb 16 2025 08:32:59
%S A113661 1,2,1,-1,0,2,2,2,1,0,0,-1,2,4,0,-1,0,2,2,0,2,0,0,2,1,4,1,-2,0,0,2,2,
%T A113661 0,0,0,-1,2,4,2,0,0,4,2,0,0,0,0,-1,3,2,0,-2,0,2,0,4,2,0,0,0,2,4,2,-1,
%U A113661 0,0,2,0,0,0,0,2,2,4,1,-2,0,4,2,0,1,0,0,-2,0,4,0,0,0,0,4,0,2,0,0,2,2,6,0,-1,0,0,2,4,0
%N A113661 Expansion of (phi(x)^3/phi(x^3) - 1)/6 where phi() is a Ramanujan theta function.
%C A113661 Ramanujan theta functions: f(q) := Product_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).
%D A113661 Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 227, Entry 4(iv).
%H A113661 G. C. Greubel, <a href="/A113661/b113661.txt">Table of n, a(n) for n = 1..1000</a>
%H A113661 Tewodros Amdeberhan, George E. Andrews, and Roberto Tauraso, <a href="https://arxiv.org/abs/2409.20400">Further study on MacMahon-type sums of divisors</a>, arXiv:2409.20400 [math.NT], 2024. See pp. 12, 18.
%H A113661 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A113661 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A113661 a(n) is multiplicative and a(2^e) = (1-3(-1)^e)/2 if e>0, a(3^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 A113661 Moebius transform is period 12 sequence [1, 1, 0, -3, -1, 0, 1, 3, 0, -1, -1, 0, ...].
%F A113661 Expansion of ((eta(q^2)^15*eta(q^3)^2*eta(q^12)^2)/(eta(q)^6*eta(q^4)^6*eta(q^6)^5)-1)/6 in powers of q.
%F A113661 G.f.: Sum_{k>0} x^k/(1+x^k+x^(2k)) +2*x^(4k-2)/(1+x^(4k-2)+x^(8k-4)).
%F A113661 6*a(n) = A113660(n), if n>0.
%F A113661 G.f.: Sum_{k >= 1} x^k/(1 + (-x)^k + x^(2*k)). - _Peter Bala_, Jan 12 2021
%F A113661 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - _Amiram Eldar_, Nov 21 2023
%e A113661 G.f. = x + 2*x^2 + x^3 - x^4 + 2*x^6 + 2*x^7 + 2*x^8 + x^9 - x^12 + ...
%p A113661 p := x -> convert(series(add(x^n/(1+(-x)^n+x^(2*n)), n = 1..100), x, 101), polynom):
%p A113661 seq(coeff(p(x), x, n), n = 1..100); # _Peter Bala_, Jan 12 2021
%t A113661 a[n_] := SeriesCoefficient[(EllipticTheta[3, 0, q]^3/EllipticTheta[3, 0, q^3] - 1)/6, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Dec 16 2017 *)
%o A113661 (PARI) {a(n)=local(x); if(n<1, 0, x=valuation(n,2); if(n%2,1,(1-3*(-1)^x)/2)*sumdiv(n/2^x,d, kronecker(-3,d)))}
%o A113661 (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, (1-3*(-1)^e)/2, if(p==3, 1, if(p%6==1, e+1, !(e%2)))))))}
%o A113661 (PARI) {a(n)=if(n<1, 0, direuler(p=2,n, if(p==2, 2-(1-2*X)/(1-X^2), 1/(1-X)/(1-kronecker(-3,p)*X)))[n])}
%o A113661 (PARI) {a(n)=local(A); if(n<1, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)^15*eta(x^3+A)^2*eta(x^12+A)^2/ eta(x+A)^6/eta(x^4+A)^6/eta(x^6+A)^5-1)/6, n))}
%Y A113661 Cf. A093766, A113660.
%Y A113661 Cf. A000700, A000122, A010054, A121373.
%K A113661 sign,mult
%O A113661 1,2
%A A113661 _Michael Somos_, Nov 03 2005