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A113216 Triangle of polynomials P(n,x) of degree n related to Pi (see comment) and derived from Padé approximation to exp(x).

%I A113216 #14 Jul 07 2017 12:27:59
%S A113216 1,1,2,1,-6,-12,1,12,-60,-120,1,-20,-180,840,1680,1,30,-420,-3360,
%T A113216 15120,30240,1,-42,-840,10080,75600,-332640,-665280,1,56,-1512,-25200,
%U A113216 277200,1995840,-8648640,-17297280,1,-72,-2520,55440,831600,-8648640,-60540480,259459200,518918400,1,90,-3960,-110880
%N A113216 Triangle of polynomials P(n,x) of degree n related to Pi (see comment) and derived from Padé approximation to exp(x).
%C A113216 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.
%F A113216 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).
%F A113216 P(n, x) = Sum_{0<=i<=n} (-1)^floor(i/2)*(2n-i)!/i!/(n-i)!*x^i.
%e A113216 P(5,x) = x^5 + 30*x^4 - 420*x^3 - 3360*x^2 + 15120*x + 30240.
%e A113216 Triangle begins:
%e A113216 1;
%e A113216 1,2;
%e A113216 1,-6,-12;
%e A113216 1,12,-60,-120;
%e A113216 1,-20,-180,840,1680;
%e A113216 1,30,-420,-3360,15120,30240;
%e A113216 1,-42,-840,10080,75600,-332640,-665280;
%e A113216 ...
%o A113216 (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))
%o A113216 (PARI) P(n,x)=sum(i=0,n,x^i*(-1)^floor(i/2)/(n-i)!/i!*(2*n-i)!)
%Y A113216 Cf. A113025 (unsigned variant), A048854, A059344, A119274.
%K A113216 sign,tabl
%O A113216 0,3
%A A113216 _Benoit Cloitre_, Jan 07 2006