|
%I #22 Nov 14 2023 04:12:16
%S 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,
%T 0,2,-2,0,0,0,-1,2,-4,2,0,0,-4,2,0,0,0,0,-1,3,-2,0,-2,0,-2,0,-4,2,0,0,
%U 0,2,-4,2,-1,0,0,2,0,0,0,0,-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 Expansion of (1-phi(x^3)^3/phi(x))/2 where phi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) := Prod_{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) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
%D Bruce C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, 1985, see p. 375, Entry 35.
%H G. C. Greubel, <a href="/A113974/b113974.txt">Table of n, a(n) for n = 1..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F a(n) is multiplicative and a(2^e) = ((-1)^e-3)/2, 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). [corrected by _Amiram Eldar_, Nov 14 2023]
%F Moebius transform is period 12 sequence [1, -3, 0, 1, -1, 0, 1, -1, 0, 3, -1, 0, ...].
%F G.f.: (1-theta_3(q^3)^3/theta_3(q))/2.
%F G.f.: Sum_{k>0} x^(3k-2)/(1-(-x)^(3k-2)) - x^(3k-1)/(1-(-x)^(3k-1)) = Sum_{k>0} -(-1)^k x^k/(1+x^k+x^(2k)) -2 x^(4k)/(1+x^(4k)+x^(8k)).
%F -2*a(n) = A113973(n), if n>0.
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = -Pi/(6*sqrt(3)) = -0.302299... . - _Amiram Eldar_, Nov 14 2023
%t a[n_]:= SeriesCoefficient[(1 - EllipticTheta[3, 0, q^3]^3/EllipticTheta[ 3, 0, q])/2, {q, 0, n}]; Table[a[n], {n, 1, 50}] (* _G. C. Greubel_, Dec 16 2017 *)
%t f[p_, e_] := If[Mod[p, 6] == 1, e + 1, (1 + (-1)^e)/2]; f[2, e_] := ((-1)^e - 3)/2; f[3, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 14 2023 *)
%o (PARI) {a(n)=local(x); if(n<1, 0, x=valuation(n,2); if(n%2,1,(-3+(-1)^x)/2)*sumdiv(n/2^x,d, kronecker(-3,d)))}
%o (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, (-3+(-1)^e)/2, if(p==3, 1, if(p%6==1, e+1, !(e%2)))))))}
%o (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 (PARI) {a(n)=local(A); if(n<0, 0, A=sum(k=1,sqrtint(n), 2*x^k^2, 1+x*O(x^n)); polcoeff( (1-subst(A+x*O(x^(n\3)),x,x^3)^3/A)/2, n))}
%Y Cf. A000700, A000122, A010054, A113973, A121373.
%K sign,easy,mult
%O 1,2
%A _Michael Somos_, Nov 10 2005
|
