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A259655
Expansion of psi(x^2) * f(-x^3)^3 / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.
5
1, 1, 3, 1, 4, 1, 5, 2, 5, 2, 5, 3, 7, 1, 7, 2, 9, 3, 7, 2, 6, 4, 11, 3, 8, 3, 10, 3, 8, 4, 9, 3, 14, 2, 10, 2, 15, 6, 7, 5, 7, 3, 14, 5, 14, 3, 16, 5, 8, 4, 13, 5, 13, 3, 12, 4, 18, 5, 14, 4, 13, 5, 15, 4, 15, 5, 16, 7, 9, 6, 11, 7, 22, 3, 16, 3, 19, 7, 16
OFFSET
0,3
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of psi(x^2) * c(x) / (3 * x^(1/3)) in powers of x where psi() is a Ramanujan theta function and c() is a cubic AGM function.
Expansion of f(-x^3)^3 / (chi(-x) * chi(-x^2)^2) in powers of x where chi(), f() are Ramanujan theta functions.
Expansion of q^(-7/12) * eta(q^3)^3 * eta(q^4)^2 / (eta(q) * eta(q^2)) in powers of q.
Euler transform of period 12 sequence [ 1, 2, -2, 0, 1, -1, 1, 0, -2, 2, 1, -3, ...].
G.f.: Product_{k>0} (1 + x^k) * (1 + x^(2*k))^2 * (1 - x^(3*k))^3.
a(2*n) = A185220(n). a(2*n + 1) = A181648(n).
EXAMPLE
G.f. = 1 + x + 3*x^2 + x^3 + 4*x^4 + x^5 + 5*x^6 + 2*x^7 + 5*x^8 + ...
G.f. = q^7 + q^19 + 3*q^31 + q^43 + 4*q^55 + q^67 + 5*q^79 + 2*q^91 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ -x^2, x^2]^2 QPochhammer[ x^3]^3, {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^3 EllipticTheta[ 2, 0, x] / (2 x^(1/4) QPochhammer[ x]), {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^4 + A)^2 / (eta(x + A) * eta(x^2 + A)), n))};
CROSSREFS
Sequence in context: A364098 A363521 A166050 * A221185 A242746 A363258
KEYWORD
nonn
AUTHOR
Michael Somos, Jul 02 2015
STATUS
approved