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

Author

Michael Somos, Jul 06 2004

Keywords

• nonn
• easy

Comments

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

Examples

			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 + ...
		

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).

Crossrefs

Cf. A033686, A116607, A281722.

Cf. A004016, A005882, A005928.

Programs

Magma

A := Basis( ModularForms( Gamma0(9), 2), 61); A[1] + 3*A[2] + 9*A[3]; /* Michael Somos, Aug 25 2014 */

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 *)

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)};

Formula

G.f.: Product_{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