A113974 Expansion of (1-phi(x^3)^3/phi(x))/2 where phi() is a Ramanujan theta function.

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, 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, 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

Offset

1,2

Comments

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).

References

Bruce C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, 1985, see p. 375, Entry 35.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

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]

Moebius transform is period 12 sequence [1, -3, 0, 1, -1, 0, 1, -1, 0, 3, -1, 0, ...].

G.f.: (1-theta_3(q^3)^3/theta_3(q))/2.

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)).

-2*a(n) = A113973(n), if n>0.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = -Pi/(6*sqrt(3)) = -0.302299... . - Amiram Eldar, Nov 14 2023

Mathematica

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 *)
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 *)

Prog

(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)))}
(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)))))))}
(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])}
(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))}

Crossrefs

Cf. A000700, A000122, A010054, A113973, A121373.

Sequence in context: A353366 A122860 A113661 * A123331 A235141 A347386

Adjacent sequences: A113971 A113972 A113973 * A113975 A113976 A113977

Keyword

sign,easy,mult

Author

Michael Somos, Nov 10 2005

Status

approved