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A233034
Expansion of (f(-x^2) / phi(-x^3))^2 in powers of x where phi(), f() are Ramanujan theta functions.
3
1, 0, -2, 4, -1, -8, 14, -4, -23, 40, -10, -60, 98, -24, -140, 224, -54, -304, 478, -112, -627, 968, -224, -1236, 1884, -432, -2346, 3540, -801, -4320, 6454, -1448, -7742, 11472, -2556, -13548, 19936, -4408, -23226, 33952, -7462, -39080, 56800, -12416, -64660
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 q^(-2/3) * b(q^2) * c(q^2) / (3 * f(-q^3)^4) in powers of q where b(), c() are cubic AGM theta functions.
Expansion of q^(-1/6) * (eta(q^2) * eta(q^6) / eta(q^3)^2)^2 in powers of q.
Euler transform of period 6 sequence [ 0, -2, 4, -2, 0, 0, ...].
G.f.: Product_{k>0} ( (1 - x^(2*k)) * (1 - x^(6*k)) / (1 - x^(3*k))^2 )^2.
a(n) = A092848(2*n) = A128111(2*n) = A182057(4*n) = A062242(4*n + 1) = A182056(4*n + 1) = A139032(6*n + 1) = A164615(6*n + 1) = A182033(6*n + 1) = A058531(12*n + 2) = A093073(12*n + 2) = A128143(12*n + 2) = A128145(12*n + 2) = A143840(12*n + 2) = A182032(12*n + 2) = A193261(12*n + 2).
-a(n) = A062244(4*n + 1) = A182034(6*n + 1) = A182035(6*n + 1) = A128144(12*n + 2) = A132976(12*n + 3) = A164268(12*n + 2) = A164612(12*n + 3) = A182035(12*n + 2).
EXAMPLE
G.f. = 1 - 2*x^2 + 4*x^3 - x^4 - 8*x^5 + 14*x^6 - 4*x^7 - 23*x^8 + 40*x^9 + ...
G.f. = q - 2*q^13 + 4*q^19 - q^25 - 8*q^31 + 14*q^37 - 4*q^43 - 23*q^49 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2] QPochhammer[ x^6] / QPochhammer[ x^3]^2)^2, {x, 0, n}];
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A) / eta(x^3 + A)^2)^2, n))};
KEYWORD
sign
AUTHOR
Michael Somos, Dec 03 2013
STATUS
approved