A132218 Expansion of psi(-x^3) / phi(-x) in powers of x where psi(), phi() are Ramanujan theta functions.

1, 2, 4, 7, 12, 20, 32, 50, 76, 113, 166, 240, 342, 482, 672, 928, 1270, 1724, 2323, 3108, 4132, 5460, 7174, 9376, 12192, 15780, 20332, 26086, 33334, 42432, 53817, 68018, 85680, 107584, 134674, 168092, 209210, 259680, 321484, 396996, 489056, 601052, 737024

Offset

0,2

Comments

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

Links

Table of n, a(n) for n=0..42.

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-3/8) * eta(q^2) * eta(q^3) * eta(q^12) / (eta(q)^2 * eta(q^6)) in powers of q.

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

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

a(n) ~ 5^(1/4) * exp(sqrt(5*n/6)*Pi) / (2^(11/4) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015

Example

G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 12*x^4 + 20*x^5 + 32*x^6 + 50*x^7 + 76*x^8 + ...
G.f. = q^3 + 2*q^11 + 4*q^19 + 7*q^27 + 12*q^35 + 20*q^43 + 32*q^51 + 50*q^59 + ...

Mathematica

nmax=60; CoefficientList[Series[Product[(1+x^k) * (1-x^(12*k))/( (1-x^k) * (1+x^(3*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)
a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/8) EllipticTheta[ 2, Pi/4, x^(3/2)] / EllipticTheta[ 4, 0, x], {x, 0, n}]; (* Michael Somos, Nov 01 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^12 + A) / (eta(x + A)^2 * eta(x^6 + A)), n))};

Crossrefs

Sequence in context: A122515 A193840 A036372 * A101230 A128129 A014968

Adjacent sequences: A132215 A132216 A132217 * A132219 A132220 A132221

Keyword

nonn

Author

Michael Somos, Aug 13 2007

Status

approved