A096726 Expansion of eta(q^3)^10 / (eta(q) * eta(q^9))^3 in powers of q.

1, 3, 9, 12, 21, 18, 36, 24, 45, 12, 54, 36, 84, 42, 72, 72, 93, 54, 36, 60, 126, 96, 108, 72, 180, 93, 126, 12, 168, 90, 216, 96, 189, 144, 162, 144, 84, 114, 180, 168, 270, 126, 288, 132, 252, 72, 216, 144, 372, 171, 279, 216, 294, 162, 36, 216, 360, 240, 270, 180, 504

Offset

0,2

Comments

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, 1991, see p. 475, Entry 7(i).

Links

Antti Karttunen, Table of n, a(n) for n = 0..16384

Bruce C. Berndt, Song Heng Chan, Zhi‐Guo Liu, and Hamza Yesilyurt, A new identity for (q;q)10 [inf] with an application to Ramanujan's partition congruence modulo 11, Quart. J. of Math., 55 (2004), pp. 13-30; alternative link.

J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. MR1010408 (91e:33012).

Formula

G.f. Prod_{k>0} (1 - x^(3*k))^10 / ((1 - x^k) * (1 - x^(9*k)))^3 = 1 + Sum_{k>0} k * (3*x^k / (1 - x^k) - 27 * x^(9*k) / (1 - x^(9*k))).

Euler transform of period 9 sequence [ 3, 3, -7, 3, 3, -7, 3, 3, -4, ...].

a(n) = 3 * b(n) where b(n) is multiplicative and b(3^e) = 1 + 3*(e>0), b(p^e) = (p^(e+1) - 1) / (p - 1) otherwise.

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2*w + 4*u*w^2 + v^3 - 6*u*v*w.

Expansion of b(q^3)^3 / b(q) = c(q)^3 / (9*c(q^3)) = (a(q)^2 + 3*a(q^3)^2) / 4 = (a(q)^2 + a(q)*b(q) + b(q)^2) / 3 in powers of q where a(), b(), c() are cubic AGM theta functions.

G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 9 (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Aug 25 2014

a(3*n + 2) = A281722(3*n + 2) + 27 * A033686(n). a(n) == A281722(n) (mod 27). - Michael Somos, Sep 04 2017

Sum_{k=1..n} a(k) ~ c * n^2, where c = 2*Pi^2/9 = 2.193245... . - Amiram Eldar, Dec 28 2023

Example

G.f. = 1 + 3*x + 9*x^2 + 12*x^3 + 21*x^4 + 18*x^5 + 36*x^6 + 24*x^7 + 45*x^8 + ...

Mathematica

CoefficientList[ Series[1 + Sum[k(3x^k/(1 - x^k) - 27x^(9k)/(1 - x^(9k))), {k, 1, 60}], {x, 0, 60}], x] (* Robert G. Wilson v, Jul 14 2004 *)
a[ n_] := If[ n < 1, Boole[ n == 0], 3 Sum[ If[ Mod[ d, 9] > 0, d, 0], {d, Divisors @ n }]]; (* Michael Somos, Aug 25 2014 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q^3]^10 / (QPochhammer[ q] QPochhammer[ q^9])^3, {q, 0, n}]; (* Michael Somos, Aug 25 2014 *)

Prog

(PARI) {a(n) = if( n<1, n==0, 3 * sigma(n) - if( n%9==0, 27 * sigma(n/9)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^10 / (eta(x + A) * eta(x^9 + A))^3, n))};
(PARI) {a(n) = polcoeff( sum(k=1, n, k*3* (x^k / (1 - x^k) - 9*x^(9*k) / (1 - x^(9*k))), 1 + x * O(x^n)), n)};
(Magma) A := Basis( ModularForms( Gamma0(9), 2), 61); A[1] + 3*A[2] + 9*A[3]; /* Michael Somos, Aug 25 2014 */

Crossrefs

Cf. A033686, A116607, A281722.

Cf. A004016, A005882, A005928.

Sequence in context: A309394 A285564 A140979 * A272027 A310323 A212059

Adjacent sequences: A096723 A096724 A096725 * A096727 A096728 A096729

Keyword

nonn,easy

Author

Michael Somos, Jul 06 2004

Status

approved