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A113448
Expansion of (eta(q^2)^2 * eta(q^9) * eta(q^18)) / (eta(q) * eta(q^6)) in powers of q.
4
1, 1, 0, 1, 0, 0, 2, 1, 0, 0, 0, 0, 2, 2, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 1, 2, 0, 2, 0, 0, 2, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, 3, 0, 1, 0, 0, 2, 2, 0
OFFSET
1,7
COMMENTS
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
LINKS
FORMULA
Euler transform of period 18 sequence [ 1, -1, 1, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 1, -1, 1, -2, ...].
Moebius transform is period 18 sequence [ 1, 0, -1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 1, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = 1, a(3^e) = 0^e, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
a(3*n) = 0, a(2*n) = a(n).
G.f.: Sum_{k>0} x^(6*k - 5) / (1 - x^(6*k - 5)) - x^(6*k - 1) / (1 - x^(6*k - 1)) - x^(18*k - 15) / (1 - x^(18*k - 15)) + x^(18*k - 6) / (1 - x^(18*k - 6)).
G.f.: Sum_{k>0} x^k * (1 - x^(2*k)) * (1 - x^(4*k)) * (1-x^(10*k)) / (1 - x^(18*k)).
Expansion of (c(q) + c(q^2))/3 in powers of q^(1/3) where c(q) is a cubic AGM theta function.
a(3*n + 1) = A033687(n). a(6*n + 1) = A097195(n). - Michael Somos, Jul 30 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(3*sqrt(3)) = 0.604599... (A073010). - Amiram Eldar, Oct 15 2022
EXAMPLE
G.f. = x + x^2 + x^4 + 2*x^7 + x^8 + 2*x^13 + 2*x^14 + x^16 + 2*x^19 + ...
MATHEMATICA
a[ n_] := If[ n < 1, 0, If[ Mod[n, 3] == 0, 0, DivisorSum[ n, KroneckerSymbol[ -12, #] &]]]; (* Michael Somos, Jul 30 2015 *)
a[ n_] := SeriesCoefficient[ x QPochhammer[ x^9]^3 / QPochhammer[ x^3] + x^2 QPochhammer[ x^18]^3 / QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Jul 30 2015 *)
PROG
(PARI) {a(n) = if( n<1, 0, if( n%3, sumdiv(n, d, kronecker(-12, d))))};
(PARI) {a(n) = if( n<1, 0, direuler(p=2, n, if( p==3, 1, 1 / ((1 - X) * (1 - kronecker(-12, p)*X))))[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%6==1, e+1, !(e%2))))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^9 + A) * eta(x^18 + A) / (eta(x + A) * eta(x^6 + A)), n))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^9 + A)^3 / eta(x^3 + A) + x * eta(x^18 + A)^3 / eta(x^6 + A), n))};
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Michael Somos, Nov 02 2005
STATUS
approved