A267852 Expansion of psi(x) * psi(x^9) * f(-x^3) / psi(x^3)^2 in powers of x where psi(), and f() are Ramanujan theta functions.

1, 1, 0, -2, -3, 0, 2, 4, 0, -5, -5, 0, 9, 8, 0, -12, -14, 0, 16, 20, 0, -23, -25, 0, 36, 37, 0, -47, -54, 0, 60, 71, 0, -84, -91, 0, 115, 121, 0, -149, -164, 0, 188, 210, 0, -245, -264, 0, 321, 343, 0, -406, -443, 0, 505, 554, 0, -641, -687, 0, 813, 863, 0

Offset

0,4

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 13, 10th equation.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 f(-x) * f(-x^6) * f(-x^3, -x^15) / f(-x, -x^5)^2 in powers of x where f(, ) is Ramanujan's general theta function.

Expansion of q^(-5/8) * eta(q^2)^2 * eta(q^3)^3 * eta(q^18)^2 / (eta(q) * eta(q^6)^4 * eta(q^9)) in powers of q.

Euler transform of period 18 sequence [ 1, -1, -2, -1, 1, 0, 1, -1, -1, -1, 1, 0, 1, -1, -2, -1, 1, -1, ...].

a(3*n) = A262614(n). a(3*n + 1) = A263041(n). a(3*n + 2) = 0.

Example

G.f. = 1 + x - 2*x^3 - 3*x^4 + 2*x^6 + 4*x^7 - 5*x^9 - 5*x^10 + 9*x^12 + ...
G.f. = q^5 + q^13 - 2*q^29 - 3*q^37 + 2*q^53 + 4*q^61 - 5*q^77 - 5*q^85 + ...

Mathematica

a[ n_] := SeriesCoefficient[ x^(-1/2) QPochhammer[ x^3] EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^(9/2)] / EllipticTheta[ 2, 0, x^(3/2)]^2, {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^3 * eta(x^18 + A)^2 / (eta(x + A) * eta(x^6 + A)^4 * eta(x^9 + A)), n))};

Crossrefs

Cf. A053269, A262614, A263041.

Sequence in context: A126045 A024307 A369287 * A328568 A219864 A257844

Adjacent sequences: A267849 A267850 A267851 * A267853 A267854 A267855

Keyword

sign

Author

Michael Somos, Apr 17 2016

Status

approved