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A279929 Expansion of c(q)*c(q^2)/9 - c(q^3)*c(q^6)/3 in powers of q where c() is a cubic AGM theta function. 1
1, 1, 0, 1, 6, 0, 8, 1, 0, 6, 12, 0, 14, 8, 0, 1, 18, 0, 20, 6, 0, 12, 24, 0, 31, 14, 0, 8, 30, 0, 32, 1, 0, 18, 48, 0, 38, 20, 0, 6, 42, 0, 44, 12, 0, 24, 48, 0, 57, 31, 0, 14, 54, 0, 72, 8, 0, 30, 60, 0, 62, 32, 0, 1, 84, 0, 68, 18, 0, 48, 72, 0, 74, 38, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..5000

FORMULA

Expansion of (2*a(q)^2 - a(q)*a(q^2) - 4*a(q^2)^2 + 3*a(q^3)*a(q^6)) / 18 in powers of q where a() is a cubic AGM theta function.

Expansion of (eta(q^3) * eta(q^6))^3 / (eta(q) * eta(q^2)) - 3 * (eta(q^9) * eta(q^18))^3 / (eta(q^3) * eta(q^6)) in powers of q.

a(n) is multiplicative with a(2^e) = 1, a(3^e) = 0^e, a(p^e) = (p^(e+1) - 1) / (p-1) if p>3.

G.f. is a period 1 Fourier series that satisfies f(-1 / (18 t)) = 2 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A281786.

a(2*n) = a(n). a(3*n) = 0.

EXAMPLE

G.f. = q + q^2 + q^4 + 6*q^5 + 8*q^7 + q^8 + 6*q^10 + 12*q^11 + 14*q^13 + ...

MATHEMATICA

a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 3, 1, # == 3, 0, True, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger @ n)];

PROG

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^6 + A))^3 / (eta(x + A) * eta(x^2 + A)) - 3 * x^2 * (eta(x^9 + A) * eta(x^18 + A))^3 / (eta(x^3 + A) * eta(x^6 + A)), 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, 1, p==3, 0, (p^(e+1) - 1) / (p - 1))))};

(MAGMA) A := Basis( ModularForms( Gamma0(18), 2), 75); A[2] +A[3] +A[5] +6*A[6];

CROSSREFS

Cf. A281786.

Sequence in context: A201521 A011393 A066362 * A244812 A083680 A271869

Adjacent sequences:  A279926 A279927 A279928 * A279930 A279931 A279932

KEYWORD

nonn,mult

AUTHOR

Michael Somos, Apr 11 2017

STATUS

approved

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Last modified September 18 01:54 EDT 2021. Contains 347504 sequences. (Running on oeis4.)