A263073 Expansion of phi(-x^5) / (chi(-x) * chi(-x^15)) in powers of x where phi(), chi() are Ramanujan theta functions.

1, 1, 1, 2, 2, 1, 2, 3, 2, 4, 4, 4, 5, 6, 6, 8, 9, 9, 12, 12, 13, 16, 18, 18, 22, 24, 25, 29, 32, 34, 40, 43, 45, 52, 56, 60, 68, 74, 78, 88, 95, 101, 113, 122, 130, 145, 156, 166, 184, 198, 209, 231, 249, 264, 290, 311, 331, 361, 388, 412, 448, 480, 510, 554

Offset

0,4

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^(-2/3) * eta(q^2) * eta(q^5)^2 * eta(q^30) / (eta(q) * eta(q^10) * eta(q^15)) in powers of q.

Euler transform of period 30 sequence [1, 0, 1, 0, -1, 0, 1, 0, 1, -1, 1, 0, 1, 0, 0, 0, 1, 0, 1, -1, 1, 0, 1, 0, -1, 0, 1, 0, 1, -1, ...].

a(n) ~ exp(sqrt(7*n/5)*Pi/3) / (2*sqrt(5*n)). - Vaclav Kotesovec, Jul 11 2016

Example

G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + x^5 + 2*x^6 + 3*x^7 + 2*x^8 + 4*x^9 + ...
G.f. = q^2 + q^5 + q^8 + 2*q^11 + 2*q^14 + q^17 + 2*q^20 + 3*q^23 + 2*q^26 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^5] / (QPochhammer[ x, x^2] QPochhammer[ x^15, x^30]), {x, 0, n}];
nmax = 100; CoefficientList[Series[Product[(1+x^k) * (1-x^(5*k)) * (1+x^(15*k)) / (1+x^(5*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^2 + A) * eta(x^5 + A)^2 * eta(x^30 + A) / (eta(x + A) * eta(x^10 + A) * eta(x^15 + A)), n))};
(PARI) q='q+O('q^99); Vec(eta(q^2)*eta(q^5)^2*eta(q^30)/(eta(q)*eta(q^10)*eta(q^15))) \\ Altug Alkan, Jul 31 2018

Crossrefs

Sequence in context: A264401 A173304 A029251 * A133091 A112204 A129710

Adjacent sequences: A263070 A263071 A263072 * A263074 A263075 A263076

Keyword

nonn

Author

Michael Somos, Oct 08 2015

Status

approved