A134415 Expansion of phi(x) / f(-x)^6 in powers of x where phi(), f() are Ramanujan theta functions..

1, 8, 39, 152, 513, 1560, 4382, 11552, 28899, 69168, 159372, 355224, 768885, 1621296, 3339201, 6732232, 13311450, 25854744, 49398043, 92953016, 172451760, 315744072, 570997539, 1020691248, 1804730732, 3158323272, 5473566645, 9398873032, 15998363307, 27005721648

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(1/4) * eta(q^2)^5 / (eta(q)^8 * eta(q^4)^2) in powers of q.

Euler transform of period 4 sequence [ 8, 3, 8, 5, ...].

G.f.: Product_{k>0} (1 + x^k)^3 / ((1 - x^k)^5 * (1 + x^(2*k))^2).

a(n) = A134414(4*n - 1).

a(n) ~ exp(2*Pi*sqrt(n)) / (16*n^2). - Vaclav Kotesovec, Sep 08 2015

Convolution inverse of A244276. - Michael Somos, Oct 25 2015

Example

G.f. = 1 + 8*x + 39*x^2 + 152*x^3 + 513*x^4 + 1560*x^5 + 4382*x^6 + 11552*x^7 + ...
G.f. = 1/q + 8*q^3 + 39*q^7 + 152*q^11 + 513*q^15 + 1560*q^19 + 4382*q^23 + ...

Mathematica

nmax = 40; CoefficientList[Series[Product[(1 + x^k)^3 / ((1 - x^k)^5 * (1 + x^(2*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] / QPochhammer[ x]^6, {x, 0, n}]; (* Michael Somos, Oct 25 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / (eta(x + A)^8 * eta(x^4 + A)^2), n))};

Crossrefs

Cf. A134414.

Sequence in context: A063002 A055581 A297017 * A097787 A215731 A144414

Adjacent sequences: A134412 A134413 A134414 * A134416 A134417 A134418

Keyword

nonn

Author

Michael Somos, Oct 26 2007

Status

approved