%I A113216
%S A113216 1,1,2,1,6,12,1,12,60,120,1,20,180,840,1680,1,30,420,3360,15120,30240,
               1,42,
%T A113216 840,10080,75600,332640,665280,1,56,1512,25200,277200,1995840,8648640,
               17297280,
%U A113216 1,72,2520,55440,831600,8648640,60540480,259459200,518918400,1,90,3960,
               110880
%V A113216 1,1,2,1,-6,-12,1,12,-60,-120,1,-20,-180,840,1680,1,30,-420,-3360,15120,
               30240,1,-42,
%W A113216 -840,10080,75600,-332640,-665280,1,56,-1512,-25200,277200,1995840,-8648640,
               -17297280,
%X A113216 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 Pade 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....) . It is relevant to note that 
               P(n,-Pi/2)^2/(r(n)-Pi/2) --> infinity 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)
%e A113216 P(5,x)=x^5 + 30*x^4 - 420*x^3 - 3360*x^2 + 15120*x + 30240
%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))
%Y A113216 Sequence in context: A049949 A106192 A113025 this_sequence A081064 A128534 
               A002562
%Y A113216 Adjacent sequences: A113213 A113214 A113215 this_sequence A113217 A113218 
               A113219
%K A113216 sign,tabl
%O A113216 0,3
%A A113216 Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 07 2006