cp's OEIS Frontend

A113448 Expansion of (eta(q^2)^2 * eta(q^9) * eta(q^18)) / (eta(q) * eta(q^6)) in powers of q.

%I A113448 #22 Nov 20 2024 16:17:27
%S A113448 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,
%T A113448 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,
%U A113448 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
%N A113448 Expansion of (eta(q^2)^2 * eta(q^9) * eta(q^18)) / (eta(q) * eta(q^6)) in powers of q.
%C A113448 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A113448 G. C. Greubel, <a href="/A113448/b113448.txt">Table of n, a(n) for n = 1..1000</a>
%F A113448 Euler transform of period 18 sequence [ 1, -1, 1, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 1, -1, 1, -2, ...].
%F A113448 Moebius transform is period 18 sequence [ 1, 0, -1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 1, 0, -1, 0, ...].
%F A113448 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).
%F A113448 a(3*n) = 0, a(2*n) = a(n).
%F A113448 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)).
%F A113448 G.f.: Sum_{k>0} x^k * (1 - x^(2*k)) * (1 - x^(4*k)) * (1-x^(10*k)) / (1 - x^(18*k)).
%F A113448 Expansion of (c(q) + c(q^2))/3 in powers of q^(1/3) where c(q) is a cubic AGM theta function.
%F A113448 a(3*n + 1) = A033687(n). a(6*n + 1) = A097195(n). - _Michael Somos_, Jul 30 2015
%F A113448 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
%e A113448 G.f. = x + x^2 + x^4 + 2*x^7 + x^8 + 2*x^13 + 2*x^14 + x^16 + 2*x^19 + ...
%t A113448 a[ n_] := If[ n < 1, 0, If[ Mod[n, 3] == 0, 0, DivisorSum[ n, KroneckerSymbol[ -12, #] &]]]; (* _Michael Somos_, Jul 30 2015 *)
%t A113448 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 *)
%o A113448 (PARI) {a(n) = if( n<1, 0, if( n%3, sumdiv(n,d, kronecker(-12, d))))};
%o A113448 (PARI) {a(n) = if( n<1, 0, direuler(p=2, n, if( p==3, 1, 1 / ((1 - X) * (1 - kronecker(-12, p)*X))))[n])}
%o A113448 (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))))};
%o A113448 (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))};
%o A113448 (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))};
%Y A113448 Cf. A004016, A005882, A005928, A033687, A073010, A097195.
%K A113448 nonn,mult
%O A113448 1,7
%A A113448 _Michael Somos_, Nov 02 2005