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A328789 Expansion of (chi(x^3) / chi(-x^2))^2 in powers of x where chi() is a Ramanujan theta function. 5
1, 0, 2, 2, 3, 4, 7, 6, 11, 14, 17, 22, 32, 34, 49, 60, 72, 90, 117, 132, 171, 206, 245, 298, 369, 422, 522, 620, 728, 868, 1043, 1198, 1439, 1688, 1962, 2304, 2717, 3114, 3668, 4258, 4909, 5698, 6627, 7566, 8788, 10112, 11574, 13310, 15317, 17410, 20010 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution square of A097242.
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 1/2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A328795.
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(1/12) * (eta(q^4) * eta(q^6)^2)^2 / (eta(q^2) * eta(q^3) * eta(q^12))^2 in powers of q.
Euler transform of period 12 sequence [0, 2, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, ...].
G.f.: Product_{k>=1} (1 + x^(6*k - 3))^2 / (1 - x^(4*k - 2))^2.
a(n) = A112206(2*n).
a(n) ~ exp(2*Pi*sqrt(n)/3) / (4*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Oct 31 2019
EXAMPLE
G.f. = 1 + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 7*x^6 + 6*x^7 + 11*x^8 + ...
G.f. = q^-1 + 2*q^23 + 2*q^35 + 3*q^47 + 4*q^59 + 7*q^71 + 6*q^83 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^2, x^2] QPochhammer[ -x^3, x^6])^2, {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A) * eta(x^6 + A)^2)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A))^2, n))};
CROSSREFS
Sequence in context: A088633 A213042 A114952 * A086969 A014692 A058670
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 27 2019
STATUS
approved

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Last modified May 26 05:37 EDT 2024. Contains 372807 sequences. (Running on oeis4.)