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A113216
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Triangle of polynomials P(n,x) of degree n related to Pi (see comment) and derived from Pade approximation to exp(x).
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0
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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, 1, 56, -1512, -25200, 277200, 1995840, -8648640, -17297280, 1, -72, -2520, 55440, 831600, -8648640, -60540480, 259459200, 518918400, 1, 90, -3960, -110880
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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.
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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
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EXAMPLE
| P(5,x)=x^5 + 30*x^4 - 420*x^3 - 3360*x^2 + 15120*x + 30240
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;
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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)!)
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CROSSREFS
| Cf. A113025 (unsigned variant), A048854, A059344, A119274.
Sequence in context: A049949 A106192 A113025 * A081064 A128534 A002562
Adjacent sequences: A113213 A113214 A113215 * A113217 A113218 A113219
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KEYWORD
| sign,tabl
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AUTHOR
| Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 07 2006
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