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A144859
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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 v_n(x), used to approximate x->sin(Pi*x)/Pi.
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2
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0, 1, -1, 1, -10, 3, 1, -140, 21, -10, 1, -3360, 1638, -360, 35, 1, -25872, 63756, -2970, 385, -126, 1, -7303296, 720720, -845988, 23023, -9828, 462, 1, -80995200, 39969072, -65739960, 1286285, -114660, 6930, -1716, 1, -57839907840
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| All even coefficients of v_n are 0. Sum_{k=0..n} T(n,k) = 0. 1/v(n)(1/2) is an approximation to Pi. D(v_n)(0) = 1 if n>0.
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LINKS
| Alois P. Heinz, Rows n = 0..99, flattened
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FORMULA
| See program.
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EXAMPLE
| 0, 1, -1, 1, -10/7, 3/7, 1, -140/87, 21/29, -10/87, 1, -3360/2047, 1638/2047, -360/2047, 35/2047, 1, -25872/15731, 63756/78655, -2970/15731, 385/15731, -126/78655 ... = A144859/A144860
As triangle:
0
1, -1
1, -10/7, 3/7
1, -140/87, 21/29, -10/87
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MAPLE
| v:= 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, `if`(n=0, NULL, D(f)(0)=1), seq((D@@i)(f)(1)=-(D@@i)(f)(0), i=2..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 (v(n)(x), x, 2*k+1): seq (seq (numer (T(n, k)), k=0..n), n=0..9);
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CROSSREFS
| Denominators of T(n, k): A144860. Diagonal gives: A110556(n) for n>0 and (-1)^n A001700(n-1) for n>0. First column gives: A057427. Cf. A144846.
Sequence in context: A178643 A038304 A159005 * A010172 A087869 A167764
Adjacent sequences: A144856 A144857 A144858 * A144860 A144861 A144862
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KEYWORD
| frac,sign,tabl
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AUTHOR
| Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 23 2008
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