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A260514 Expansion of phi(x) * chi(x^2)^4 in powers of x where phi(), chi() are Ramanujan theta functions. 2
1, 2, 4, 8, 8, 12, 16, 16, 29, 36, 44, 64, 72, 88, 112, 128, 162, 202, 244, 304, 352, 420, 496, 576, 703, 820, 968, 1152, 1320, 1544, 1792, 2048, 2405, 2782, 3204, 3728, 4240, 4856, 5568, 6320, 7259, 8276, 9416, 10752, 12144, 13760, 15568, 17536, 19875, 22416 (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..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^(1/3) * eta(q^2) * eta(q^4)^6 / (eta(q)^2 * eta(q^8)^4) in powers of q.

Euler transform of period 8 sequence [ 2, 1, 2, -5, 2, 1, 2, -1, ...].

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

EXAMPLE

G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 8*x^4 + 12*x^5 + 16*x^6 + 16*x^7 + ...

G.f. = 1/q + 2*q^2 + 4*q^5 + 8*q^8 + 8*q^11 + 12*q^14 + 16*q^17 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^2, x^4]^4, {x, 0, n}];

nmax=60; CoefficientList[Series[Product[(1+x^k) * (1-x^(4*k))^6 / ((1-x^k) * (1-x^(8*k))^4), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^6 / (eta(x + A)^2 * eta(x^8 + A)^4), n))};

(PARI) q='q+O('q^99); Vec(eta(q^2)*eta(q^4)^6/(eta(q)^2*eta(q^8)^4)) \\ Altug Alkan, Aug 01 2018

CROSSREFS

Sequence in context: A054785 A236924 A266575 * A123263 A008218 A073043

Adjacent sequences:  A260511 A260512 A260513 * A260515 A260516 A260517

KEYWORD

nonn

AUTHOR

Michael Somos, Jul 27 2015

STATUS

approved

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Last modified August 7 17:02 EDT 2020. Contains 336277 sequences. (Running on oeis4.)