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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.

%I A144846 #30 Jul 19 2023 18:20:28
%S A144846 0,1,-1,7,-5,3,87,-35,63,-5,2047,-105,819,-45,35,78655,-8085,15939,
%T A144846 -7425,1925,-63,4439935,-57057,225225,-211497,115115,-2457,231,
%U A144846 344674687,-4429425,17486469,-8217495,9003995,-200655,24255,-429
%N 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.
%C A144846 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).
%H A144846 Alois P. Heinz, <a href="/A144846/b144846.txt">Rows n = 0..99, flattened</a>
%H A144846 Alois P. Heinz, <a href="/A144846/a144846.gif">Animation of Pi*u_n(x) for n=0..15, x=-3..3</a>
%F A144846 See program.
%e A144846 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
%e A144846 As triangle:
%e A144846    0;
%e A144846    1/2,   -1/2;
%e A144846    7/8,   -5/4,   3/8;
%e A144846   87/88, -35/22, 63/88, -5/44;
%e A144846   ...
%p A144846 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);
%t A144846 f[x_] := Sum[a[2i+1] x^(2i+1), {i, 0, n}];
%t A144846 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 A144846 T[n_, k_] := Coefficient[u[n][x], x, 2k+1];
%t A144846 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 *)
%Y A144846 Denominators of T(n,k): A144847.
%Y A144846 Diagonal gives: (-1)^n A001790(n) for n>1.
%Y A144846 Cf. A000796.
%K A144846 frac,sign,tabl,look
%O A144846 0,4
%A A144846 _Alois P. Heinz_, Sep 22 2008