A277582 Expansion of f(-x^5) * f(-x^7) - x * f(-x) * f(-x^35) in powers of q where f() is a Ramanujan theta function.

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

Offset

0,36

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..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1) * (eta(q^10) * eta(q^14) - eta(q^2) * eta(q^70)) in powers of q^2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (140 t)) = 140^(1/2) (t/i) f(t) where q = exp(2 Pi i t).

Example

G.f. = 1 - x + x^2 + x^3 - x^5 - x^6 - x^7 - x^8 - x^10 + x^12 + x^13 + ...
G.f. = q - q^3 + q^5 + q^7 - q^11 - q^13 - q^15 - q^17 - q^21 + q^25 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^5] QPochhammer[ x^7] - x QPochhammer[ x] QPochhammer[ x^35], {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^5 + A) * eta(x^7 + A) - x * eta(x + A) * eta(x^35 + A), n))};
(PARI) {a(n) = my(j=24*n+12, n1, n2, t1, t2); if( n<0, 0, n1 = sqrtint(j\7); n2 = sqrtint(j\35); sum(k = (-n1+4)\6, (n1-1)\6, if( (t1 = j-7*(6*k+1)^2)%5 ==0 && issquare( t1/5, &t2), if( t2%6 == 5, t2=-t2); m = (t2-1)/6; (-1)^(k+m))) - sum(k = (-n2+4)\6, (n2-1)\6, t1 = j-35*(6*k+1)^2; if( issquare( t1, &t2), if( t2%6 == 5, t2=-t2); m = (t2-1)/6; (-1)^(k+m))))};

Crossrefs

Cf. A030216.

Sequence in context: A330231 A323017 A273638 * A037803 A184318 A030410

Adjacent sequences: A277579 A277580 A277581 * A277583 A277584 A277585

Keyword

sign

Author

Michael Somos, Oct 21 2016

Status

approved