A260514 Expansion of phi(x) * chi(x^2)^4 in powers of x where phi(), chi() are Ramanujan theta functions.

1, 2, 4, 8, 8, 12, 16, 16, 29, 36, 44, 64, 72, 88, 112, 128, 162, 202, 244, 304, 352, 420, 496, 576, 703, 820, 968, 1152, 1320, 1544, 1792, 2048, 2405, 2782, 3204, 3728, 4240, 4856, 5568, 6320, 7259, 8276, 9416, 10752, 12144, 13760, 15568, 17536, 19875, 22416

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..2500

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^(1/3) * eta(q^2) * eta(q^4)^6 / (eta(q)^2 * eta(q^8)^4) in powers of q.

Euler transform of period 8 sequence [ 2, 1, 2, -5, 2, 1, 2, -1, ...].

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

Example

G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 8*x^4 + 12*x^5 + 16*x^6 + 16*x^7 + ...
G.f. = 1/q + 2*q^2 + 4*q^5 + 8*q^8 + 8*q^11 + 12*q^14 + 16*q^17 + ...

Mathematica

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

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^6 / (eta(x + A)^2 * eta(x^8 + A)^4), n))};
(PARI) q='q+O('q^99); Vec(eta(q^2)*eta(q^4)^6/(eta(q)^2*eta(q^8)^4)) \\ Altug Alkan, Aug 01 2018

Crossrefs

Sequence in context: A054785 A236924 A266575 * A123263 A008218 A073043

Adjacent sequences: A260511 A260512 A260513 * A260515 A260516 A260517

Keyword

nonn

Author

Michael Somos, Jul 27 2015

Status

approved