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, 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

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 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. A132974, A132979.

Sequence in context: A282522 A329523 A331317 * A321880 A075673 A062827

Adjacent sequences: A259661 A259662 A259663 * A259665 A259666 A259667

Keyword

nonn

Author

Michael Somos, Jul 02 2015

Status

approved