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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
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
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
KEYWORD
nonn,easy,mult
AUTHOR
Michael Somos, Aug 21 2005, Apr 18 2007
STATUS
approved