A233034 Expansion of (f(-x^2) / phi(-x^3))^2 in powers of x where phi(), f() are Ramanujan theta functions.

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

Author

Michael Somos, Dec 03 2013

Keywords

• sign

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).

Examples

			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 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A058531, A062242, A062244, A092848, A093073, A128111, A128143, A128144, A128145, A132976, A139032, A143840, A164268, A164612, A164615, A182032, A182033, A182034, A182035, A182056, A182057, A193261.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2] QPochhammer[ x^6] / QPochhammer[ x^3]^2)^2, {x, 0, n}];

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))};

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).