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A260599 Expansion of psi(x^4) / chi(-x)^2 in powers of x where psi(), chi() are Ramanujan theta functions. 1
1, 2, 3, 6, 10, 16, 25, 38, 55, 80, 115, 160, 223, 306, 415, 560, 747, 988, 1301, 1700, 2206, 2850, 3661, 4676, 5950, 7536, 9500, 11936, 14936, 18620, 23141, 28662, 35386, 43566, 53480, 65466, 79937, 97356, 118277, 143370, 173391, 209232, 251966, 302806 (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 f(-x^4)^4 / (f(-x)^2 * phi(x^2)) in powers of x where phi(), f() are Ramanujan  theta functions.

Expansion of q^(-7/12) * eta(q^2)^2 * eta(q^8)^2 / (eta(q)^2 * eta(q^4)) in powers of q.

Euler transform of period 8 sequence [ 2, 0, 2, 1, 2, 0, 2, -1, ...].

2 * a(n) = A260574(4*n + 2).

a(n) ~ exp(sqrt(2*n/3)*Pi) / (8*sqrt(2*n)). - Vaclav Kotesovec, Oct 14 2015

EXAMPLE

G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 25*x^6 + 38*x^7 + ...

G.f. = q^7 + 2*q^19 + 3*q^31 + 6*q^43 + 10*q^55 + 16*q^67 + 25*q^79 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x]^2 EllipticTheta[ 2, 0, x^2] / (2 x^(1/2)), {x, 0, n}];

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

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

CROSSREFS

Cf. A260574.

Sequence in context: A075623 A024801 A324742 * A280908 A146163 A101277

Adjacent sequences:  A260596 A260597 A260598 * A260600 A260601 A260602

KEYWORD

nonn

AUTHOR

Michael Somos, Jul 29 2015

STATUS

approved

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Last modified January 27 04:57 EST 2020. Contains 331291 sequences. (Running on oeis4.)