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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.

%I A144859 #21 Jul 19 2023 18:25:21
%S A144859 0,1,-1,1,-10,3,1,-140,21,-10,1,-3360,1638,-360,35,1,-25872,63756,
%T A144859 -2970,385,-126,1,-7303296,720720,-845988,23023,-9828,462,1,-80995200,
%U A144859 39969072,-65739960,1286285,-114660,6930,-1716,1,-57839907840
%N 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.
%C A144859 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.
%H A144859 Alois P. Heinz, <a href="/A144859/b144859.txt">Rows n = 0..99, flattened</a>
%H A144859 Alois P. Heinz, <a href="/A144859/a144859.gif">Animation of Pi*v_n(x) for n=0..15, x=-3..3</a>
%F A144859 See program.
%e A144859 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
%e A144859 As triangle:
%e A144859   0
%e A144859   1,   -1
%e A144859   1,  -10/7,   3/7
%e A144859   1, -140/87, 21/29, -10/87
%p A144859 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);
%t A144859 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 *)
%Y A144859 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.
%K A144859 frac,sign,tabl,look
%O A144859 0,5
%A A144859 _Alois P. Heinz_, Sep 23 2008