A144846 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.

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

Offset: 0

Author

Alois P. Heinz, Sep 22 2008

Keywords

• frac
• sign
• tabl
• look

Comments

All even coefficients of u_n are 0. Sum_{k=0..n} T(n,k) = 0. 1/u(n)(1/2) = A230142(n)/A230143(n) is an approximation to Pi: Pi-1/u(10)(1/2) = 0.6883935...*10^(-9), Pi-1/u(50)(1/2) = 0.2993600...*10^(-47).

Examples

			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;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..99, flattened
• Alois P. Heinz, Animation of Pi*u_n(x) for n=0..15, x=-3..3

Crossrefs

Denominators of T(n,k): A144847.

Diagonal gives: (-1)^n A001790(n) for n>1.

Cf. A000796.

Programs

Maple

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);

Mathematica

f[x_] := Sum[a[2i+1] x^(2i+1), {i, 0, n}];
u[n_] := u[n] = Function[x, f[x] /. Solve[Join[{f[1] == 0}, Table[(D[f[x], {x, i}] /. x -> 1) == If[i == 1, -1, -(D[f[x], {x, i}] /. x -> 0)], {i, 1, n}]]][[1]]];
T[n_, k_] := Coefficient[u[n][x], x, 2k+1];
Table[Numerator[T[n, k]], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 12 2014, translated from Maple, updated May 31 2016 *)

Formula

See program.