A131944 Expansion of (1 - b(q)*b(q^2)) / 3 where b() is a cubic AGM function. Expansion of (1 - eta(q)^3 * eta(q^2)^3 / (eta(q^3) * eta(q^6))) / 3 in powers of q.

1, 1, -5, 1, 6, -5, 8, 1, -23, 6, 12, -5, 14, 8, -30, 1, 18, -23, 20, 6, -40, 12, 24, -5, 31, 14, -77, 8, 30, -30, 32, 1, -60, 18, 48, -23, 38, 20, -70, 6, 42, -40, 44, 12, -138, 24, 48, -5, 57, 31, -90, 14, 54, -77, 72, 8, -100, 30, 60, -30, 62, 32, -184

Offset

1,3

Comments

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

References

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 84, Eq. (32.65).

Links

Table of n, a(n) for n=1..63.

Formula

Expansion of (1 - eta(q)^3 * eta(q^2)^3 / (eta(q^3) * eta(q^6))) / 3 in powers of q.

a(n) is multiplicative with a(2^e) = 1, a(3^e) = 4- 3^(e+1), a(p^e) = (p^(e+1) - 1) / (p-1) if p>3.

G.f.: (1 - Product_{k>0} ((1 - x^k) * (1 -x^(2*k)))^3 / ((1 - x^(3*k)) * (1 - x^(6*k)))) / 3.

G.f.: Sum_{k>0} (6*k-1) * x^(6*k-1) / (1 - x^(6*k-1)) - 2 * (6*k-5) * x^(6*k-3) / (1 - x^(6*k-3)) + (6*k-5) * x^(6*k-5) / (1 - x^(6*k-5)).

-3 * a(n) = A131943(n) unless n=0.

Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = 4*Pi^2/81 = 0.487387... . - Amiram Eldar, Jan 22 2024

Example

G.f. = q + q^2 - 5*q^3 + q^4 + 6*q^5 - 5*q^6 + 8*q^7 + q^8 - 23*q^9 + 6*q^10 +...

Mathematica

a[ n_] := If[ n < 1, 0, Sum[ d {0, 1, 0, -2, 0, 1}[[ Mod[ d, 6] + 1]], {d, Divisors @ n}]]; (* Michael Somos, Nov 11 2015 *)
QP = QPochhammer; s = (1 - QP[q]^3*(QP[q^2]^3/(QP[q^3]*QP[q^6])))/(3 q) + O[q]^70; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d*((d%6==1) + (d%6==5) - 2*(d%6==3))))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^3 * eta(x^2 + A)^3 / (eta(x^3 + A) * eta(x^6 + A)))/3, 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, 4 - p^(e+1), (p^(e+1) - 1) / (p-1) )))};

Crossrefs

Cf. A004016, A005882, A005928, A131943.

Sequence in context: A318265 A318553 A176909 * A228475 A296355 A306700

Adjacent sequences: A131941 A131942 A131943 * A131945 A131946 A131947

Keyword

sign,mult

Author

Michael Somos, Jul 30 2007

Status

approved