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A263073 Expansion of phi(-x^5) / (chi(-x) * chi(-x^15)) in powers of x where phi(), chi() are Ramanujan theta functions. 2
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 (list; graph; refs; listen; history; text; internal format)
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.

M. 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,changed

AUTHOR

Michael Somos, Oct 08 2015

STATUS

approved

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Last modified November 15 18:59 EST 2019. Contains 329149 sequences. (Running on oeis4.)