A101195 Expansion of psi(x^3) / psi(x) in powers of x where psi() is a Ramanujan theta function.

1, -1, 1, -1, 2, -3, 3, -4, 6, -7, 8, -10, 13, -16, 18, -22, 28, -33, 38, -45, 55, -65, 74, -87, 104, -121, 138, -160, 188, -217, 247, -284, 330, -378, 428, -489, 562, -640, 722, -820, 936, -1059, 1191, -1345, 1524, -1717, 1924, -2163, 2438, -2734, 3054, -3419

Offset

0,5

Comments

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

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

Formula

Expansion of q^(-1/2) * theta_2(q^3) / theta_2(q) in powers of q^2.

Expansion of q^(-1/4) * eta(q) * eta(q^6)^2 / (eta(q^2)^2 * eta(q^3)) in powers of q.

Given g.f. A(x), B(q) = A(q^4) * q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = u^3 - v + 3*u*v^2 - 3*u^2*v^3.

Given g.f. A(x), B(q) = (A(q^4) * q)^2 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2 - v + v^2 + 3*u^2*v.

a(n) = (-1)^n * A036018(n).

G.f.: Product_{k>0} (1 + x^k + x^(2*k)) * (1 - x^k + x^(2*k))^2.

a(n) ~ (-1)^n * exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

Example

G.f. = 1 - x + x^2 - x^3 + 2*x^4 - 3*x^5 + 3*x^6 - 4*x^7 + 6*x^8 + ...
G.f. = q - q^5 + q^9 - q^13 + 2*q^17 - 3*q^21 + 3*q^25 - 4*q^29 + 6*q^33 + ...

Mathematica

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

Prog

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

Crossrefs

Cf. A036018.

Sequence in context: A029033 A041003 A067592 * A036018 A123552 A071610

Adjacent sequences: A101192 A101193 A101194 * A101196 A101197 A101198

Keyword

sign

Author

Michael Somos, Dec 03 2004

Status

approved