A111932 Expansion of q * (psi(q) * psi(q^3))^2 in powers of q where psi() is a Ramanujan theta function.

1, 2, 1, 4, 6, 2, 8, 8, 1, 12, 12, 4, 14, 16, 6, 16, 18, 2, 20, 24, 8, 24, 24, 8, 31, 28, 1, 32, 30, 12, 32, 32, 12, 36, 48, 4, 38, 40, 14, 48, 42, 16, 44, 48, 6, 48, 48, 16, 57, 62, 18, 56, 54, 2, 72, 64, 20, 60, 60, 24, 62, 64, 8, 64, 84, 24, 68, 72, 24, 96, 72, 8, 74, 76, 31

Offset

1,2

Comments

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

References

Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 223 Entry 3(iii).

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 87, Eq. (33.2).

Links

Seiichi Manyama, 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

Expansion of (1/3) * (b(q^2)^2 / b(q))* (c(q^2)^2 / c(q)) in powers of q where b(), c() are cubic AGM theta functions.

Expansion of (eta(q^2) * eta(q^6))^4 / (eta(q) * eta(q^3))^2 in powers of q.

Euler transform of period 6 sequence [ 2, -2, 4, -2, 2, -4, ...].

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

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u*w * (u - 4*v) - v * (v - 4*w)^2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (3/4) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A131946. - Michael Somos, Sep 19 2013

G.f.: Sum_{k>0} k * x^k * (1 - x^(2*k))^2 / (1 - x^(6*k)) = x * Product_{k>0} ((1 + x^k) * (1 + x^(3*k)))^4 * ((1 - x^k) * (1 - x^(3*k)))^2.

a(3*n) = a(n), a(2*n) = 2 * a(n).

Convolution square of A033762. - Michael Somos, Sep 19 2013

From Amiram Eldar, Sep 12 2023: (Start)

Dirichlet g.f.: (1 - 1/2^s) * (1 - 1/3^(s-1)) * zeta(s-1) * zeta(s).

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/24 = 0.411233... (A222171). (End)

Example

G.f. = q + 2*q^2 + q^3 + 4*q^4 + 6*q^5 + 2*q^6 + 8*q^7 + 8*q^8 + q^9 + ...

Mathematica

a[ n_] := If[ n < 1, 0, Sum[ Mod[n/d, 2] d KroneckerSymbol[ 9, d], { d, Divisors[ n]}]]; (* Michael Somos, Sep 19 2013 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^2] QPochhammer[ q^6])^4 / (QPochhammer[ q] QPochhammer[ q^3])^2, {q, 0, n}]; (* Michael Somos, Sep 19 2013 *)

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, (n/d % 2) * d * (d%3>0)))};
(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, p^e, if( p==3, 1, (p^(e+1) - 1) / (p-1)))))) };
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^4 / (eta(x + A) * eta(x^3 + A))^2, n))};
(Sage) A = ModularForms( Gamma0(6), 2, prec=50) . basis();  A[1] + 2*A[2]; # Michael Somos, Sep 19 2013

Crossrefs

Cf. A033762, A131946, A222171.

Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

Sequence in context: A120769 A187141 A165604 * A121456 A323286 A193818

Adjacent sequences: A111929 A111930 A111931 * A111933 A111934 A111935

Keyword

nonn,easy,mult

Author

Michael Somos, Aug 21 2005, Apr 18 2007

Status

approved