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A102315 Expansion of (b(q^6) * c(q^6)) / (b(q^3) * c(q^3)) in powers of q where b(), c() are cubic AGM theta functions. 1
1, 2, 3, 8, 13, 20, 37, 56, 83, 134, 196, 280, 419, 592, 824, 1176, 1618, 2202, 3040, 4096, 5471, 7368, 9753, 12824, 16937, 22090, 28653, 37248, 47968, 61488, 78887, 100472, 127461, 161702, 203951, 256368, 322090, 402748, 502112, 625464, 776061 (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

FORMULA

Expansion of (chi(-x) * chi(-x^3))^(-2) in powers of x where chi() is a Ramanujan theta function.

Euler transform of period 6 sequence [2, 0, 4, 0, 2, 0, ...].

Given g.f. A(x), then B(q) = q*A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2 - v - 4*u*v^2.

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

Convolution inverse of A058543. - Michael Somos, Feb 19 2015

a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(11/4)*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 08 2015

EXAMPLE

G.f. = 1 + 2*x + 3*x^2 + 8*x^3 + 13*x^4 + 20*x^5 + 37*x^6 + 56*x^7 + ...

G.f. = q + 2*q^4 + 3*q^7 + 8*q^10 + 13*q^13 + 20*q^16 + 37*q^19 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ (QPochhammer[ x, x^2] QPochhammer[ x^3, x^6])^-2, {x, 0, n}]; (* Michael Somos, Feb 19 2015 *)

nmax = 60; CoefficientList[Series[Product[(1+x^k)^2 * (1+x^(3*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 08 2015 *)

PROG

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

(PARI) q='q+O('q^99); Vec((eta(q^2)*eta(q^6)/(eta(q)*eta(q^3)))^2) \\ Altug Alkan, Apr 21 2018

CROSSREFS

Cf. A058543.

Sequence in context: A236169 A080478 A002053 * A142880 A147329 A175148

Adjacent sequences:  A102312 A102313 A102314 * A102316 A102317 A102318

KEYWORD

nonn

AUTHOR

Michael Somos, Jan 04 2005

STATUS

approved

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Last modified June 26 10:12 EDT 2019. Contains 324375 sequences. (Running on oeis4.)