A166243 - OEIS

%I #17 Aug 04 2018 14:33:24

%S 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,

%T 0,0,-360,0,0,-540,0,0,-243,-152,0,0,-16440,0,0,2700,0,0,2430,0,0,729,

%U 0,0,24240,0,0,1311840,0,0,-17010,0,0,-10206,0,0,-2187,0,6848,0,0,-2974800

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

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

%H Seiichi Manyama, <a href="/A166243/b166243.txt">Rows n = -1..99, flattened</a>

%F 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).

%e V(-1) = 0;

%e V(0) = 1;

%e V(1) = 0 + 0*x - 3*x^2;

%e V(2) = 0 + 2*x + 0*x^2 + 0*x^3 + 9*x^4.

%o (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; }

%Y Cf. A166244, A159843, A166246, A206309.

%K sign,tabf

%O -1,5

%A _Max Alekseyev_, Oct 10 2009