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.
Original entry on oeis.org
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
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
-
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);
-
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 *)
A230144
Numerator of 1/v_n(1/2), where polynomial v_n(x) is used to approximate x->sin(Pi*x)/Pi.
Original entry on oeis.org
8, 224, 1856, 1048064, 80542720, 18185973760, 2823575035904, 4608812904218624, 398274484258471936, 766890677854431870976, 298370458295691854741504, 553395519598838736006152192, 301475731054794304317380624384, 381273851270136749855228020391936
Offset: 1
8/3, 224/75, 1856/595, 1048064/333795, 80542720/25638459, 18185973760/5788790007, 2823575035904/898772045457 ... = A230144/A230145
-
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: seq(numer(1/v(n)(1/2)), n=1..15);
-
v[n_] := v[n] = Module[{f, i, x, a}, f[x_] := Sum[a[2*i+1]*x^(2*i+1), {i, 0, n}]; Function[x, Sum[a[2*i+1]*x^(2*i+1), {i, 0, n}] /. First @ Solve[Join[{f[1] == 0}, {If[n == 0, True, f'[0] == 1]}, Table[Derivative[i][f][1] == -Derivative[i][f][0], {i, 2, n}]]]]]; Table[Numerator[1/v[n][1/2]], {n, 1, 15}] (* Jean-François Alcover, Feb 13 2014, after Maple *)
A230145
Denominator of 1/v_n(1/2), where polynomial v_n(x) is used to approximate x->sin(Pi*x)/Pi.
Original entry on oeis.org
3, 75, 595, 333795, 25638459, 5788790007, 898772045457, 1467030741832227, 126774706022852173, 244108884436744360605, 94974266622893811200463, 176151264858556860995936775, 95962705639251788100721754775, 121363236202656183485569513082175
Offset: 1
-
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: seq(denom(1/v(n)(1/2)), n=1..15);
-
v[n_] := v[n] = Module[{f, i, x, a}, f[x_] := Sum[a[2*i+1]*x^(2*i+1), {i, 0, n}]; Function[x, Sum[a[2*i+1]*x^(2*i+1), {i, 0, n}] /. First @ Solve[Join[{f[1] == 0}, {If[n == 0, True, f'[0] == 1]}, Table[Derivative[i][f][1] == -Derivative[i][f][0], {i, 2, n}]]]]]; Table[Denominator[1/v[n][1/2]], {n, 1, 15}] (* Jean-François Alcover, Feb 13 2014, after Maple *)
Showing 1-3 of 3 results.
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