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A262368
Expansion of f(x^2, x^2) * f(x, x^2)^2 in powers of x where f(, ) is Ramanujan's general theta function.
1
1, 2, 5, 6, 7, 6, 4, 8, 8, 14, 11, 8, 8, 6, 15, 14, 12, 12, 8, 14, 16, 12, 16, 10, 19, 20, 12, 14, 12, 14, 21, 14, 16, 14, 16, 30, 12, 20, 16, 20, 20, 16, 24, 12, 25, 26, 16, 16, 16, 28, 16, 14, 25, 18, 24, 30, 20, 16, 28, 38, 32, 14, 16, 22, 20, 28, 28, 16
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 phi(x^2) phi(-x^3)^2 / chi(-x)^2 in powers of x where phi(), chi() are Ramanujan theta functions.
Expansion of q^(-1/12) * eta(q^3)^4 * eta(q^4)^5 / (eta(q)^2 * eta(q^6)^2 * eta(q^8)^2) in powers of q.
Euler transform of period 24 sequence [ 2, 2, -2, -3, 2, 0, 2, -1, -2, 2, 2, -5, 2, 2, -2, -1, 2, 0, 2, -3, -2, 2, 2, -3, ...].
a(n) = A261426(4*n).
EXAMPLE
G.f. = 1 + 2*x + 5*x^2 + 6*x^3 + 7*x^4 + 6*x^5 + 4*x^6 + 8*x^7 + ...
G.f. = q + 2*q^13 + 5*q^25 + 6*q^37 + 7*q^49 + 6*q^61 + 4*q^73 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^2] EllipticTheta[ 4, 0, x^3]^2 QPochhammer[ -x, x]^2, {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^4 * eta(x^4 + A)^5 / (eta(x + A)^2 * eta(x^6 + A)^2 * eta(x^8 + A)^2), n))};
CROSSREFS
Cf. A261426.
Sequence in context: A225366 A146099 A004097 * A198547 A120678 A078384
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 02 2015
STATUS
approved