

A213625


Expansion of psi(x)^2 * phi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.


8



1, 2, 3, 6, 4, 4, 7, 2, 8, 10, 4, 10, 9, 6, 8, 10, 4, 8, 16, 8, 9, 12, 8, 12, 20, 6, 8, 10, 8, 18, 11, 12, 8, 20, 12, 8, 20, 6, 20, 26, 8, 8, 15, 10, 16, 18, 12, 16, 20, 10, 16, 16, 8, 24, 24, 8, 21, 26, 8, 20, 20, 14, 8, 28, 16, 10, 28, 10, 24, 22, 8, 16, 17
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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
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions


FORMULA

Expansion of q^(1/4) * eta(q^2)^2 * eta(q^4)^5 / (eta(q)^2 * eta(q^8)^2) in powers of q.
Euler transform of period 8 sequence [ 2, 0, 2, 5, 2, 0, 2, 3, ...].
G.f. is a period 1 Fourier series which satisfies f(1 / (32 t)) = 2^(3/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A116597.
a(2*n) = A213622(n). a(2*n + 1) = 2 * A132969(n).


EXAMPLE

G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 4*x^4 + 4*x^5 + 7*x^6 + 2*x^7 + 8*x^8 + 10*x^9 + ...
G.f. = q + 2*q^5 + 3*q^9 + 6*q^13 + 4*q^17 + 4*q^21 + 7*q^25 + 2*q^29 + 8*q^33 + ...


MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(1/2)]^2 EllipticTheta[ 3, 0, x^2] / (4 x^(1/4)), {x, 0, n}];


PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^4 + A)^5 / (eta(x + A)^2 * eta(x^8 + A)^2), n))};


CROSSREFS

Cf. A116597, A132969, A213622.
Sequence in context: A003573 A111804 A246835 * A132368 A157248 A085515
Adjacent sequences: A213622 A213623 A213624 * A213626 A213627 A213628


KEYWORD

nonn


AUTHOR

Michael Somos, Jun 16 2012


STATUS

approved



