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Search: id:A113216
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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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1
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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; listen)
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OFFSET
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0,3
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COMMENT
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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.
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FORMULA
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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)
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EXAMPLE
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P(5,x)=x^5 + 30*x^4 - 420*x^3 - 3360*x^2 + 15120*x + 30240
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PROGRAM
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(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))
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CROSSREFS
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Sequence in context: A049949 A106192 A113025 this_sequence A081064 A128534 A002562
Adjacent sequences: A113213 A113214 A113215 this_sequence A113217 A113218 A113219
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KEYWORD
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sign,tabl
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AUTHOR
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Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 07 2006
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