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A261446 Expansion of f(-x^3, -x^3) * f(-x, -x^5) / f(-x, -x)^2 in powers of x where f(,) is Ramanujan's general theta function. 1
1, 3, 8, 18, 38, 75, 140, 252, 439, 744, 1232, 1998, 3182, 4986, 7700, 11736, 17673, 26322, 38808, 56682, 82070, 117867, 167996, 237744, 334202, 466836, 648224, 895014, 1229148, 1679436, 2283568, 3090672, 4164578, 5587941, 7467464, 9940482, 13183238, 17421288 (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).

LINKS

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

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of f(-x^2) * f(-x^3) * f(-x^6) / f(-x)^3 in powers of x where f() is a Ramanujan theta function.

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

Euler transform of period 6 sequence [ 3, 2, 2, 2, 3, 0, ...].

a(n) = (-1)^n * A261325(n). 2 * a(2*n) = A261240(3*n + 1). a(2*n + 1) = 3 * A233698(n).

2 * a(n) = A058647(3*n + 1) = A139213(3*n + 1) = A186964(3*n + 1) = A187020(3*n + 1).

a(n) = A123649(3*n + 1) = A139214(3*n + 1) = A233693(3*n + 1).

Convolution inverse is A132301.

a(n) ~ exp(2*Pi*sqrt(n/3)) / (4 * 3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015

EXAMPLE

G.f. = 1 + 3*x + 8*x^2 + 18*x^3 + 38*x^4 + 75*x^5 + 140*x^6 + 252*x^7 + ...

G.f. = q + 3*q^4 + 8*q^7 + 18*q^10 + 38*q^13 + 75*q^16 + 140*q^19 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^3] QPochhammer[ x^6] / QPochhammer[ x]^3, {x, 0, n}];

nmax=60; CoefficientList[Series[Product[(1-x^(2*k)) * (1-x^(3*k)) * (1-x^(6*k)) / (1-x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)

PROG

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

CROSSREFS

Cf. A058647, A123649, A132301, A139213, A139214, A186964, A187020, A233693, A233698, A261240, A261325.

Sequence in context: A036635 A000713 A261325 * A078409 A036642 A000235

Adjacent sequences:  A261443 A261444 A261445 * A261447 A261448 A261449

KEYWORD

nonn

AUTHOR

Michael Somos, Aug 19 2015

STATUS

approved

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Last modified May 19 10:36 EDT 2019. Contains 323390 sequences. (Running on oeis4.)