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A111932 Expansion of q * (psi(q) * psi(q^3))^2 in powers of q where psi() is a Ramanujan theta function. 6
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 (list; graph; refs; listen; history; text; internal format)
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

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

N. 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

M. 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

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.

Sequence in context: A120769 A187141 A165604 * A121456 A323286 A193818

Adjacent sequences:  A111929 A111930 A111931 * A111933 A111934 A111935

KEYWORD

nonn,mult

AUTHOR

Michael Somos, Aug 21 2005, Apr 18 2007

STATUS

approved

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Last modified July 9 14:41 EDT 2020. Contains 335543 sequences. (Running on oeis4.)