A144859 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.

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

Offset

0,5

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, cf. A230144/A230145. D(v_n)(0) = 1 if n>0.

Links

Alois P. Heinz, Rows n = 0..99, flattened

Alois P. Heinz, Animation of Pi*v_n(x) for n=0..15, x=-3..3

Formula

See program.

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

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);

Mathematica

v[n_] := v[n] = Module[{f, i, x, a}, f[x_] = Sum[a[2*i+1]*x^(2i+1), {i, 0, n}]; Function[x, Sum[a[2*i+1]*x^(2i+1), {i, 0, n}] /. First @ Solve [{f[1] == 0, If[n == 0, True, f'[0] == 1], Sequence @@ Table[Derivative[i][f][1] == -Derivative[i][f][0], {i, 2, n}]}, Table[a[2*i+1], {i, 0, n}]]]]; T[n_, k_] := Coefficient[v[n][x], x, 2*k+1]; Table[Table[Numerator[T[n, k]], {k, 0, n}], {n, 0, 9}] // Flatten (* Jean-François Alcover, Feb 12 2014, translated from Maple *)

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: A354941 A352689 A159005 * A280519 A010172 A370632

Adjacent sequences: A144856 A144857 A144858 * A144860 A144861 A144862

Keyword

frac,sign,tabl,look

Author

Alois P. Heinz, Sep 23 2008

Status

approved