A113661 Expansion of (phi(x)^3/phi(x^3) - 1)/6 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) := 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).

References

Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 227, Entry 4(iv).

Links

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

Tewodros Amdeberhan, George E. Andrews, and Roberto Tauraso, Further study on MacMahon-type sums of divisors, arXiv:2409.20400 [math.NT], 2024. See pp. 12, 18.

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) = (1-3(-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).

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

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.

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

6*a(n) = A113660(n), if n>0.

G.f.: Sum_{k >= 1} x^k/(1 + (-x)^k + x^(2*k)). - Peter Bala, Jan 12 2021

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

Example

G.f. = x + 2*x^2 + x^3 - x^4 + 2*x^6 + 2*x^7 + 2*x^8 + x^9 - x^12 + ...

Maple

p := x -> convert(series(add(x^n/(1+(-x)^n+x^(2*n)), n = 1..100), x, 101), polynom):
seq(coeff(p(x), x, n), n = 1..100); # Peter Bala, Jan 12 2021

Mathematica

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 *)

Prog

(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)))}
(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)))))))}
(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<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))}

Crossrefs

Cf. A093766, A113660.

Cf. A000700, A000122, A010054, A121373.

Sequence in context: A058394 A353366 A122860 * A113974 A123331 A235141

Adjacent sequences: A113658 A113659 A113660 * A113662 A113663 A113664

Keyword

sign,mult

Author

Michael Somos, Nov 03 2005

Status

approved