A213267 Expansion of phi(q^9) / (psi(-q) * chi(q^3)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 12, 15, 20, 26, 32, 39, 50, 63, 76, 92, 114, 140, 168, 201, 244, 295, 350, 415, 496, 591, 696, 818, 967, 1140, 1332, 1554, 1820, 2126, 2468, 2861, 3324, 3855, 4448, 5126, 5916, 6816, 7824, 8970, 10292, 11793, 13471, 15372, 17548

Offset

0,5

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

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 eta(q^2) * eta(q^3) * eta(q^12) * eta(q^18)^5 / (eta(q) * eta(q^4) * eta(q^6)^2 * eta(q^9)^2 * eta(q^36)^2) in powers of q.

Euler transform of period 36 sequence [ 1, 0, 0, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 0, 0, 1, 1, -2, 1, 1, 0, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 0, 0, 1, 0, ...].

a(n) = A132975(n) unless n=0.

a(2*n) = A128129(n). a(2*n + 1) = A132302.

a(3*n) = A164617(n). a(3*n + 1) = A132977(n). a(3*n + 2) = A132978(n).

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

Example

1 + q + q^2 + q^3 + 2*q^4 + 3*q^5 + 4*q^6 + 5*q^7 + 7*q^8 + 10*q^9 + ...

Mathematica

nmax=60; CoefficientList[Series[Product[(1+x^k) * (1+x^(6*k)) * (1+x^(9*k))^5 * (1-x^(9*k))^3 / ((1-x^(4*k)) * (1+x^(3*k)) * (1-x^(36*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)

Prog

(PARI) {a(n) = local(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^18 + A)^5 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2 * eta(x^9 + A)^2 * eta(x^36 + A)^2), n))}

Crossrefs

Cf. A128129, A132302, A132975, A132977, A132978, A164617.

Sequence in context: A036033 A124243 A132975 * A145977 A050729 A117536

Adjacent sequences: A213264 A213265 A213266 * A213268 A213269 A213270

Keyword

nonn

Author

Michael Somos, Jun 07 2012

Status

approved