A113973 - OEIS

%I #25 Nov 14 2023 04:25:45

%S 1,-2,4,-2,2,0,4,-4,4,-2,0,0,2,-4,8,0,2,0,4,-4,0,-4,0,0,4,-2,8,-2,4,0,

%T 0,-4,4,0,0,0,2,-4,8,-4,0,0,8,-4,0,0,0,0,2,-6,4,0,4,0,4,0,8,-4,0,0,0,

%U -4,8,-4,2,0,0,-4,0,0,0,0,4,-4,8,-2,4,0,8,-4,0,-2,0,0,4,0,8,0,0,0,0,-8,0,-4,0,0,4,-4,12,0,2,0,0,-4,8

%N Expansion of phi(x^3)^3/phi(x) 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="/A113973/b113973.txt">Table of n, a(n) for n = 0..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) = -2*b(n) where b(n) is multiplicative and b(2^e) = (1-3(-1)^e)/2 if e>0, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).

%F Euler transform of period 12 sequence [ -2, 3, 4, 1, -2, -6, -2, 1, 4, 3, -2, -2, ...].

%F Moebius transform is period 12 sequence [ -2, 6, 0, -2, 2, 0, -2, 2, 0, -6, 2, 0, ...].

%F Expansion of (eta(q)^2*eta(q^4)^2*eta(q^6)^15)/ (eta(q^2)^5*eta(q^3)^6*eta(q^12)^6) in powers of q.

%F G.f.: theta_3(q^3)^3/theta_3(q).

%F G.f.: 1+2( Sum_{k>0} x^(3k-1)/(1-(-x)^(3k-1)) - x^(3k-2)/(1-(-x)^(3k-2))) = 1 +2( Sum_{k>0} (-1)^k x^k/(1+x^k+x^(2k)) +2 x^(4k)/(1+x^(4k)+x^(8k)) ).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(3*sqrt(3)) = 0.604599... (A073010). - _Amiram Eldar_, Nov 14 2023

%t s = EllipticTheta[3, 0, q^3]^3/EllipticTheta[3, 0, q] + O[q]^105; CoefficientList[s, q] (* _Jean-François Alcover_, Dec 04 2015 *)

%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[0] = 1; a[1] = -2; a[n_] := -2 * Times @@ f @@@ FactorInteger[n]; Array[a, 100, 0] (* _Amiram Eldar_, Nov 14 2023 *)

%o (PARI) {a(n)=local(x); if(n<1, n==0, x=valuation(n,2); if(n%2,-2,(3-(-1)^x))*sumdiv(n/2^x,d, kronecker(-3,d)))}

%o (PARI) {a(n)=local(A,p,e); if(n<1, n==0, A=factor(n); -2*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, n==0, -2*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( subst(A+x*O(x^(n\3)),x,x^3)^3/A, n))}

%o (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)^2*eta(x^4+A)^2*eta(x^6+A)^15/ eta(x^2+A)^5/eta(x^3+A)^6/eta(x^12+A)^6, n))}

%Y a(n)=-2*A113974(n) if n>0.

%Y Cf. A000700, A000122, A010054, A073010, A121373.

%K sign,easy

%O 0,2

%A _Michael Somos_, Nov 10 2005