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A260515 Expansion of phi(x^2) * chi(x)^4 in powers of x where phi(), chi() are Ramanujan theta functions. 1
1, 4, 8, 16, 29, 44, 72, 112, 162, 244, 352, 496, 703, 968, 1320, 1792, 2405, 3204, 4240, 5568, 7259, 9416, 12144, 15568, 19875, 25260, 31944, 40256, 50523, 63140, 78672, 97680, 120870, 149148, 183480, 225056, 275350, 335984, 408920, 496544, 601514, 727044 (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 q^(1/6) * eta(q^2)^6 * eta(q^4) / (eta(q)^4 * eta(q^8)^2) in powers of q.

Euler transform of period 8 sequence [ 4, -2, 4, -3, 4, -2, 4, -1, ...].

G.f. is a period 1 Fourier series which satisifes f(-1 / (144 t)) = 6^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A260514.

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

EXAMPLE

G.f. = 1 + 4*x + 8*x^2 + 16*x^3 + 29*x^4 + 44*x^5 + 72*x^6 + 112*x^7 + ...

G.f. = 1/q + 4*q^5 + 8*q^11 + 16*q^17 + 29*q^23 + 44*q^29 + 72*q^35 + ...

MATHEMATICA

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

nmax=60; CoefficientList[Series[Product[(1-x^k) * (1+x^k)^5 / ((1+x^(2*k)) * (1+x^(4*k))^2), {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)^6 * eta(x^4 + A) / (eta(x + A)^4 * eta(x^8 + A)^2), n))};

(PARI) q='q+O('q^99); Vec(eta(q^2)^6*eta(q^4) / (eta(q)^4*eta(q^8)^2)) \\ Altug Alkan, Mar 18 2018

CROSSREFS

Cf. A260514.

Sequence in context: A271649 A128441 A009861 * A301148 A302508 A030119

Adjacent sequences:  A260512 A260513 A260514 * A260516 A260517 A260518

KEYWORD

nonn

AUTHOR

Michael Somos, Jul 27 2015

STATUS

approved

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Last modified March 19 15:02 EDT 2019. Contains 321330 sequences. (Running on oeis4.)