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A225701 Expansion of chi(q)^5 / chi(q^5) in powers of q where chi() is a Ramanujan theta function. 3
1, 5, 10, 15, 30, 55, 80, 120, 190, 285, 410, 585, 840, 1190, 1640, 2240, 3070, 4170, 5570, 7400, 9830, 12960, 16920, 21990, 28520, 36805, 47180, 60225, 76720, 97350, 122880, 154610, 194110, 242880, 302740, 376295, 466710, 577270, 711800, 875520, 1074790 (list; graph; refs; listen; history; text; internal format)
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..1000

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of eta(q^2)^10 * eta(q^5) * eta(q^20) / (eta(q)^5 * eta(q^4)^5 * eta(q^10)^2) in powers of q.

Euler transform of period 20 sequence [ 5, -5, 5, 0, 4, -5, 5, 0, 5, -4, 5, 0, 5, -5, 4, 0, 5, -5, 5, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. of A223903.

a(n) = (-1)^n * A138521(n). a(n) = 5 * A210458(n) unless n=0.

a(n) ~ exp(2*Pi*sqrt(n/5)) / (2 * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

EXAMPLE

G.f. = 1 + 5*q + 10*q^2 + 15*q^3 + 30*q^4 + 55*q^5 + 80*q^6 + 120*q^7 + 190*q^8 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2]^5 / QPochhammer[ -q^5, q^10], {q, 0, n}];

nmax=60; CoefficientList[Series[Product[(1-x^k)^5 * (1+x^k)^10 * (1+x^(10*k)) / ((1-x^(4*k))^5 * (1+x^(5*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^10 * eta(x^5 + A) * eta(x^20 + A) / (eta(x + A)^5 * eta(x^4 + A)^5 * eta(x^10 + A)^2), n))};

CROSSREFS

Cf. A138521, A210458, A223903.

Sequence in context: A031471 A045206 A138521 * A022600 A067237 A196157

Adjacent sequences:  A225698 A225699 A225700 * A225702 A225703 A225704

KEYWORD

nonn

AUTHOR

Michael Somos, May 17 2013

STATUS

approved

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Last modified July 18 07:05 EDT 2019. Contains 325134 sequences. (Running on oeis4.)