A131947 Expansion of (1 - (phi(-q) * phi(-q^3))^2)/4 in powers of q where phi() is a Ramanujan theta function.

1, -1, 1, -5, 6, -1, 8, -13, 1, -6, 12, -5, 14, -8, 6, -29, 18, -1, 20, -30, 8, -12, 24, -13, 31, -14, 1, -40, 30, -6, 32, -61, 12, -18, 48, -5, 38, -20, 14, -78, 42, -8, 44, -60, 6, -24, 48, -29, 57, -31, 18, -70, 54, -1, 72, -104, 20, -30, 60, -30, 62

Offset

1,4

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

References

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.66).

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Michael Somos, Introduction to Ramanujan theta functions.

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

Formula

a(n) is multiplicative with a(2^e) = 3 - 2^(e+1), a(3^e) = 1, a(p^e) = (p^(e+1) - 1) / (p-1) if p>3.

G.f.: Sum_{k>0} k * (-x)^k / (1 - x^k) * Kronecker(9, k) = ((theta_3(-x) * theta_3(-x^3))^2 - 1) / 4.

a(n) = -(-1)^n * A113262(n). -4 * a(n) = A131946(n) unless n=0.

Dirichlet g.f.: (1 - 1/2^(s-2)) * (1 - 1/3^(s-1)) * zeta(s-1) * zeta(s). - Amiram Eldar, Sep 12 2023

Example

G.f. = x - x^2 + x^3 - 5*x^4 + 6*x^5 - x^6 + 8*x^7 - 13*x^8 + x^9 - 6*x^10 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (1 - (EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^3])^2) / 4, {q, 0, n}]; (* Michael Somos, Nov 11 2015 *)
a[ n_] := SeriesCoefficient[ (1 - (QPochhammer[ q] QPochhammer[ q^3])^4 / (QPochhammer[ q^2] QPochhammer[ q^6])^2) / 4, {q, 0, n}]; (* Michael Somos, Nov 11 2015 *)
a[ n_] := If[ n < 1, 0, Sum[ d {0, 1, -1, 0, -1, 1}[[Mod[ d, 6] + 1]], {d, Divisors @ n}]]; (* Michael Somos, Nov 11 2015 *)
a[ n_] := If[ n < 1, 0, Sum[ n/d {6, 1, -3, -2, -3, 1}[[Mod[ d, 6] + 1]], {d, Divisors @ n}]]; (* Michael Somos, Nov 11 2015 *)

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d*((abs(d%6-3) == 2) - (abs(d%6-3) == 1))))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - (eta(x + A) * eta(x^3 + A))^4 / (eta(x^2 + A) * eta(x^6 + A))^2) / 4, n))};
(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 3 - p^(e+1), p==3, 1, (p^(e+1) - 1) / (p-1) )))};

Crossrefs

Cf. A113262, A131946.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A020798 A021182 A175647 * A113262 A195823 A105577

Adjacent sequences: A131944 A131945 A131946 * A131948 A131949 A131950

Keyword

sign,easy,mult

Author

Michael Somos, Jul 30 2007

Status

approved