A262152 Expansion of f(-x^6)^3 / (f(-x^4)^2 * psi(x)) in powers of x where phi(), f() are Ramanujan theta functions.

1, -1, 1, -2, 5, -6, 4, -8, 18, -20, 16, -27, 52, -58, 47, -74, 133, -146, 127, -187, 312, -343, 304, -431, 687, -751, 687, -941, 1436, -1569, 1459, -1948, 2879, -3139, 2975, -3885, 5569, -6071, 5826, -7472, 10457, -11385, 11067, -13972, 19122, -20813, 20423

Offset: 0

Author

Michael Somos, Sep 13 2015

Keywords

• sign

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 - x + x^2 - 2*x^3 + 5*x^4 - 6*x^5 + 4*x^6 - 8*x^7 + 18*x^8 + ...
G.f. = q^7 - q^31 + q^55 - 2*q^79 + 5*q^103 - 6*q^127 + 4*q^151 - 8*q^175 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Programs

Mathematica

a[ n_] := SeriesCoefficient[ 2 x^(1/8) QPochhammer[ x^6]^3 / (QPochhammer[ x^4]^2 EllipticTheta[ 2, 0, x^(1/2)]), {x, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^3 / (eta(x^2 + A)^2 * eta(x^4 + A)^2), n))};

PARI

q='q+O('q^99); Vec(eta(q)*eta(q^6)^3/(eta(q^2)^2*eta(q^4)^2)) \\ Altug Alkan, Jul 31 2018

Formula

Expansion of q^(-7/24) * eta(q) * eta(q^6)^3 / (eta(q^2)^2 * eta(q^4)^2) in powers of q.

Euler transform of period 12 sequence [-1, 1, -1, 3, -1, -2, -1, 3, -1, 1, -1, 0, ...].