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A096726 Expansion of eta(q^3)^10 / (eta(q) * eta(q^9))^3 in powers of q. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

REFERENCES

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 475 Entry 7(i).

LINKS

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

B. C. Berndt, S. H. Chan, Z.-G. Liu and H. 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.

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

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.

Sequence in context: A309394 A285564 A140979 * A272027 A310323 A212059

Adjacent sequences:  A096723 A096724 A096725 * A096727 A096728 A096729

KEYWORD

nonn

AUTHOR

Michael Somos, Jul 06 2004

STATUS

approved

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Last modified April 10 11:15 EDT 2021. Contains 342845 sequences. (Running on oeis4.)