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

0,2

Comments

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

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 3, 2nd equation.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

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).

Example

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 + ...

Mathematica

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

Prog

(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

Crossrefs

Cf. A053270, A256209.

Sequence in context: A193818 A127535 A285491 * A262930 A293387 A306015

Adjacent sequences: A257637 A257638 A257639 * A257641 A257642 A257643

Keyword

nonn

Author

Michael Somos, Nov 04 2015

Status

approved