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A259664 Expansion of f(-x^6) / (f(-x^2)^3 * phi(-x)^2) in powers of x where phi(), f() are Ramanujan theta functions. 1
1, 4, 15, 44, 121, 300, 707, 1572, 3366, 6932, 13865, 26952, 51187, 95080, 173280, 310172, 546438, 948360, 1623737, 2744840, 4585920, 7577684, 12393330, 20073648, 32219481, 51270912, 80927964, 126758160, 197096678, 304339020, 466829342, 711555332, 1078037580 (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

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of 1 / (b(x^2) * phi(-x)^2) in powers of x where phi() is a Ramanujan theta function and b() is a cubic AGM theta function.

Expansion of q^(-1/2) * eta(q^6)^3 / (eta(q)^4 * eta(q^2)) in powers of q.

Euler transform of period 6 sequence [ 4, 5, 4, 5, 4, 2, ...].

G.f.: Product_{k>0} (1 - x^(6*k))^3 / ((1 - x^k)^4 * (1 - x^(2*k))).

3 * a(n) = A132974(2*n + 1). -3 * a(n) = A132979(2*n + 1).

a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (6^(9/4) * n^(5/4)). - Vaclav Kotesovec, Oct 14 2015

EXAMPLE

G.f. = 1 + 4*x + 15*x^2 + 44*x^3 + 121*x^4 + 300*x^5 + 707*x^6 + 1572*x^7 + ...

G.f. = q + 4*q^3 + 15*q^5 + 44*q^7 + 121*q^9 + 300*q^11 + 707*q^13 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ x^6]^3 / (QPochhammer[ x]^4 QPochhammer[ x^2]), {x, 0, n}];

PROG

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

CROSSREFS

Cf. A132947, A132979.

Sequence in context: A294259 A240359 A282522 * A075673 A062827 A074448

Adjacent sequences:  A259661 A259662 A259663 * A259665 A259666 A259667

KEYWORD

nonn

AUTHOR

Michael Somos, Jul 02 2015

STATUS

approved

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Last modified October 19 22:28 EDT 2018. Contains 316378 sequences. (Running on oeis4.)