A257640 Expansion of psi(x)^2 / phi(-x^3) in powers of x where phi(), psi() are Ramanujan theta functions.

1, 2, 1, 4, 6, 2, 11, 14, 4, 24, 30, 10, 47, 58, 18, 88, 108, 32, 156, 188, 57, 268, 318, 94, 444, 522, 152, 716, 834, 244, 1129, 1308, 378, 1744, 2010, 576, 2652, 3038, 870, 3968, 4524, 1288, 5857, 6650, 1884, 8540, 9660, 2730, 12312, 13878, 3906, 17572

Offset: 0

Author

Michael Somos, Nov 04 2015

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 2*x + x^2 + 4*x^3 + 6*x^4 + 2*x^5 + 11*x^6 + 14*x^7 + 4*x^8 + ...
G.f. = q + 2*q^5 + q^9 + 4*q^13 + 6*q^17 + 2*q^21 + 11*q^25 + 14*q^29 + ...
		

References

• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 3, 2nd equation.

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. A053270, A256209.

Programs

Mathematica

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

PARI

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

PARI

q='q+O('q^99); Vec(eta(q^2)^4*eta(q^6)/(eta(q)^2*eta(q^3)^2)) \\ Altug Alkan, Apr 21 2018

Formula

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

Euler transform of period 6 sequence [ 2, -2, 4, -2, 2, -1, ...].

a(n) = A053270(n) unless n == 2 (mod 3). a(3*n) = A053270(3*n) - 2 * A256209(n).