login
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, 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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 13, 10th equation.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
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
KEYWORD
sign
AUTHOR
Michael Somos, Apr 17 2016
STATUS
approved