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A128639 Expansion of (1/3) * (c(q)^2 / c(q^2)) / (b(q)^2 / b(q^2)) in powers of q where b(), c() are cubic AGM theta functions. 3
1, 8, 40, 152, 488, 1392, 3640, 8896, 20584, 45512, 96816, 199200, 398072, 775216, 1475264, 2749776, 5029736, 9043344, 16005352, 27918304, 48047280, 81661504, 137183136, 227952960, 374924152, 610743224, 985891568, 1577869784, 2504850112, 3945854640, 6170415888 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

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

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of (phi(-q^3) / phi(-q))^4 in powers of q where phi() is a Ramanujan theta function.

Expansion of ((eta(q^3) / eta(q))^2 * (eta(q^2) / eta(q^6)))^4 in powers of q.

Euler transform of period 6 sequence [ 8, 4, 0, 4, 8, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u * (1-v) * (1-9*v) - (u-v)^2.

G.f.: (Product_{k>0} (1 + x^k + x^(2*k)) / (1 - x^k + x^(2*k)) )^4.

a(n) = 8 * A128638(n) unless n = 0. Convolution inverse of A128637.

a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(3/4) * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015

Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 1/3 + (1/9)*sqrt(3) + (1/9)*sqrt(9+6*sqrt(3)). - Simon Plouffe, Mar 02 2021

EXAMPLE

G.f. = 1 + 8*q + 40*q^2 + 152*q^3 + 488*q^4 + 1392*q^5 + 3640*q^6 + ...

MATHEMATICA

nmax = 40; CoefficientList[Series[Product[((1 + x^k + x^(2*k)) / (1 - x^k + x^(2*k)))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x^3 + A) / eta(x + A))^2 * eta(x^2 + A) / eta(x^6 + A))^4, n))};

CROSSREFS

Cf. A128637, A128638.

Sequence in context: A028596 A264602 A125198 * A341365 A004405 A284286

Adjacent sequences: A128636 A128637 A128638 * A128640 A128641 A128642

KEYWORD

nonn

AUTHOR

Michael Somos, Mar 16 2007

STATUS

approved

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Last modified November 30 21:57 EST 2022. Contains 358453 sequences. (Running on oeis4.)