A166243 Villegas-Zagier polynomials (listing coefficients from lowest to highest degree).

0, 1, 0, 0, -3, 0, 2, 0, 0, 9, -2, 0, 0, -18, 0, 0, -27, 0, 0, 36, 0, 0, 108, 0, 0, 81, 0, 152, 0, 0, -360, 0, 0, -540, 0, 0, -243, -152, 0, 0, -16440, 0, 0, 2700, 0, 0, 2430, 0, 0, 729, 0, 0, 24240, 0, 0, 1311840, 0, 0, -17010, 0, 0, -10206, 0, 0, -2187, 0, 6848, 0, 0, -2974800

Offset

-1,5

References

H. Cohen, Number Theory. Volume I: Tools and Diophantine Equations, Springer-Verlag, 2007, p. 378.

Links

Seiichi Manyama, Rows n = -1..99, flattened

Formula

V(-1) = 0, V(0) = 1, and for n >= 0, V(n+1) = (8*x^3-1)*V'(n) - (16*n+3)*x^2*V(n) - 4*n*(2*n-1)*x*V(n-1).

Example

V(-1) = 0;
V(0) = 1;
V(1) = 0 + 0*x - 3*x^2;
V(2) = 0 + 2*x + 0*x^2 + 0*x^3 + 9*x^4.

Prog

(PARI) { V(n) = my(p0, p1, q); if(n==-1, return(0)); p0 = 0; p1 = 1; for(m=1, n, q = (8*x^3-1)*deriv(p1) - (16*(m-1)+3)*x^2*p1 - 4*(m-1)*(2*(m-1)-1)*x*p0; p0 = p1; p1 = q; ); p1; }

Crossrefs

Cf. A166244, A159843, A166246, A206309.

Sequence in context: A067169 A257927 A011339 * A118514 A190544 A172293

Adjacent sequences: A166240 A166241 A166242 * A166244 A166245 A166246

Keyword

sign,tabf

Author

Max Alekseyev, Oct 10 2009

Status

approved