A193275 Expansion of psi(x^2) * phi(x^17) - x^4 * phi(x) * psi(x^34) in powers of x^4 where phi(), psi() are Ramanujan theta functions.

1, 0, 1, 0, -1, -2, 1, 0, -2, 0, 0, 0, 1, -2, 0, 0, 0, 2, 0, 2, -1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, -1, -2, -2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 0, 1, 0, 0, 2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 2, 0, 1, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -2, 0, -2, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0

Offset

0,6

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 (phi(q) * phi(q^68) - phi(q^4) * phi(q^17)) / (2 * q) in powers of q^4 where phi() is a Ramanujan theta function.

Expansion of q^(-1) * (eta(q^4)^2 * eta(q^34)^5 / (eta(q^2) * eta(q^17)^2 * eta(q^68)^2) - eta(q^2)^5 * eta(q^68)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^34))) in powers of q^4.

G.f. is a period 1 Fourier series which satisfies f(-1 / (272 t)) = 272^(1/2) (t/i) f(t) where q = exp(2 Pi i t).

Example

1 + x^2 - x^4 - 2*x^5 + x^6 - 2*x^8 + x^12 - 2*x^13 + 2*x^17 + 2*x^19 - ...
q + q^9 - q^17 - 2*q^21 + q^25 - 2*q^33 + q^49 - 2*q^53 + 2*q^69 + 2*q^77 - ...

Mathematica

a[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, q^(1/4)]*EllipticTheta[3, 0, q^17] - EllipticTheta[3, 0, q]*EllipticTheta[3, 0, q^(17/4)])/(2* q^(1/4)), {q, 0, n}];  Table[a[n], {n, 0, 50}] (* G. C. Greubel, Apr 03 2018 *)

Prog

(PARI) {a(n) = local(x); if( n<0, 0, n = 4*n + 1; (sum( i=1, sqrtint( n\68), issquare( n - 68*i^2)) - sum( i=1, sqrtint( (n-1)\17), issquare( n - 17*i^2, &x) && (x%2==0) )) * 2 + issquare( n) - issquare( 17*n))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 * eta(x^34 + A)^5 / (eta(x^2 + A) * eta(x^17 + A)^2 * eta(x^68 + A)^2) - x^4 * eta(x^2 + A)^5 * eta(x^68 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^34 + A)), n))}

Crossrefs

Sequence in context: A267546 A228817 A110399 * A182033 A112214 A246962

Adjacent sequences: A193272 A193273 A193274 * A193276 A193277 A193278

Keyword

sign

Author

Michael Somos, Jul 21 2011

Status

approved