OFFSET
0,3
COMMENTS
P(n,x) is a sequence of polynomials of degree n with integer coefficients, having exactly n real roots, such that r(n) the smallest root (in absolute value) converges quickly to Pi/2. e.g. the appropriate root for P(5,x) is r(5)=1.5707963(4026....) . To see the rapidity of convergence it is relevant noting that (r(n)-Pi/2)(2n)! -->0 as n grows.
FORMULA
P(0, x) = 1, P(1, x) = x+2, P(n, x) = (4*n-2)*P(n-1, x)-x^2*P(n-2, x).
P(n, x) = Sum_{0<=i<=n} (-1)^floor(i/2)*(2n-i)!/i!/(n-i)!*x^i.
EXAMPLE
P(5,x) = x^5 + 30*x^4 - 420*x^3 - 3360*x^2 + 15120*x + 30240.
Triangle begins:
1;
1,2;
1,-6,-12;
1,12,-60,-120;
1,-20,-180,840,1680;
1,30,-420,-3360,15120,30240;
1,-42,-840,10080,75600,-332640,-665280;
...
PROG
(PARI) P(n, x)=if(n<2, if(n%2, x+2, 1), (4*n-2)*P(n-1, x)-x^2*P(n-2, x))
(PARI) P(n, x)=sum(i=0, n, x^i*(-1)^floor(i/2)/(n-i)!/i!*(2*n-i)!)
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Benoit Cloitre, Jan 07 2006
STATUS
approved