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A260058
Expansion of f(x^2, x^4) * f(x^3, x^3) / f(-x, -x^2)^2 in power of x where f(, ) is Ramanujan's general theta function.
1
1, 2, 6, 14, 30, 60, 114, 208, 366, 626, 1044, 1704, 2730, 4300, 6672, 10212, 15438, 23076, 34134, 50008, 72612, 104560, 149400, 211920, 298554, 417902, 581412, 804254, 1106448, 1514316, 2062332, 2795488, 3772302, 5068632, 6782508, 9040224, 12004014, 15881692
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of eta(q^4) * eta(q^6)^7 / (eta(q)^2 * eta(q^2) * eta(q^3)^2 * eta(q^12)^3) in powers of q.
Euler transform of period 12 sequence [ 2, 3, 4, 2, 2, -2, 2, 2, 4, 3, 2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (1/6) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A187145.
a(n) = A261154(3*n).
a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*3^(5/4)*n^(3/4)). - Vaclav Kotesovec, Mar 17 2018
EXAMPLE
G.f. = 1 + 2*x + 6*x^2 + 14*x^3 + 30*x^4 + 60*x^5 + 114*x^6 + 208*x^7 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^6] QPochhammer[ -x^4, x^6] QPochhammer[ x^6] EllipticTheta[ 3, 0, x^3] / QPochhammer[ x]^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)^7 / (eta(x + A)^2 * eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^3), n))};
CROSSREFS
Cf. A261154.
Sequence in context: A366542 A192966 A339668 * A331699 A327048 A228038
KEYWORD
nonn
AUTHOR
Michael Somos, Nov 08 2015
STATUS
approved