A258327 Expansion of phi(x^3) / f(-x^2) in powers of x where phi(), f() are Ramanujan theta functions.

1, 0, 1, 2, 2, 2, 3, 4, 5, 6, 7, 10, 13, 14, 17, 22, 26, 30, 36, 44, 52, 60, 70, 84, 99, 112, 131, 156, 179, 204, 236, 274, 315, 358, 409, 472, 539, 608, 692, 792, 897, 1010, 1144, 1298, 1464, 1644, 1849, 2088, 2347, 2622, 2940, 3304, 3694, 4118, 4600, 5142

Offset

0,4

Comments

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

Links

Vaclav Kotesovec, 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 f(-x, x^2) / psi(-x) in powers of x where psi(), f() are Ramanujan theta functions.

Expansion of q^(-1/12) * eta(q^6)^5 / (eta(q^2) * eta(q^3)^2 * eta(q^12)^2) in powers of q.

Euler transform of period 12 sequence [ 0, 1, 2, 1, 0, -2, 0, 1, 2, 1, 0, 0, ...].

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

a(n) = (-1)^n * A256636(n).

a(n) ~ exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Jul 11 2016

Example

G.f. = 1 + x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + 6*x^9 + ...
G.f. = 1/q + q^23 + 2*q^35 + 2*q^47 + 2*q^59 + 3*q^71 + 4*q^83 + 5*q^95 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^3] / QPochhammer[ x^2], {x, 0, n}];
nmax = 50; CoefficientList[Series[Product[(1+x^(6*k-3)) / ((1-x^(6*k-2)) * (1-x^(6*k-3)) * (1-x^(6*k-4)) * (1+x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 11 2016 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^5 / (eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2), n))};

Crossrefs

Cf. A256636.

Sequence in context: A118301 A018121 A256636 * A102240 A026837 A366916

Adjacent sequences: A258324 A258325 A258326 * A258328 A258329 A258330

Keyword

nonn

Author

Michael Somos, May 26 2015

Status

approved