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A213267 Expansion of phi(q^9) / (psi(-q) * chi(q^3)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions. 2
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 (list; graph; refs; listen; history; text; internal format)
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

M. 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

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Last modified February 19 10:18 EST 2019. Contains 320310 sequences. (Running on oeis4.)