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Search: id:A144846
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| A144846 |
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Numerators of triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the coefficient of x^(2k+1) in polynomial u_n(x), used to approximate x->sin(Pi*x)/Pi. |
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+0
3
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| 0, 1, -1, 7, -5, 3, 87, -35, 63, -5, 2047, -105, 819, -45, 35, 78655, -8085, 15939, -7425, 1925, -63, 4439935, -57057, 225225, -211497, 115115, -2457, 231, 344674687, -4429425, 17486469, -8217495, 9003995, -200655, 24255, -429
(list; table; graph; listen)
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OFFSET
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0,4
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COMMENT
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All even coefficients of u_n are 0. Sum_{k=0..n} T(n,k) = 0. 1/u(n)(1/2) is an approximation to Pi.
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FORMULA
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See program.
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EXAMPLE
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0, 1/2, -1/2, 7/8, -5/4, 3/8, 87/88, -35/22, 63/88, -5/44, 2047/2048, -105/64, 819/1024, -45/256, 35/2048, 78655/78656, -8085/4916, 15939/19664, -7425/39328, 1925/78656, -63/39328 ... = A144846/A144847
As triangle:
0
1/2, -1/2
7/8, -5/4, 3/8
87/88, -35/22, 63/88, -5/44
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MAPLE
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u:= proc(n) option remember; local f, i, x; f:= unapply (simplify (sum ('cat (a||(2*i+1)) *x^(2*i+1)', 'i'=0..n) ), x); unapply (subs (solve ({f(1)=0, seq((D@@i)(f)(1)=`if`(i=1, -1, -(D@@i)(f)(0)), i=1..n)}, {seq (cat (a||(2*i+1)), i=0..n)}), sum ('cat (a||(2*i+1)) *x^(2*i+1)', 'i'=0..n) ), x); end: T:= (n, k)-> coeff (u(n)(x), x, 2*k+1): seq (seq (numer (T(n, k)), k=0..n), n=0..9);
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CROSSREFS
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Denominators of T(n, k): A144847. Diagonal gives: (-1)^n A001790(n) for n>1.
Adjacent sequences: A144843 A144844 A144845 this_sequence A144847 A144848 A144849
Sequence in context: A155816 A093824 A019935 this_sequence A090289 A160670 A109863
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KEYWORD
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frac,sign,tabl
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AUTHOR
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Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 22 2008
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