A212261 Array A(i,j) read by antidiagonals: A(i,j) is the (2i-1)-th derivative of sin(sin(sin(...sin(x)))) nested j times evaluated at 0.

1, 1, -1, 1, -2, 1, 1, -3, 12, -1, 1, -4, 33, -128, 1, 1, -5, 64, -731, 1872, -1, 1, -6, 105, -2160, 25857, -37600, 1, 1, -7, 156, -4765, 121600, -1311379, 990784, -1, 1, -8, 217, -8896, 368145, -10138880, 89060065, -32333824, 1

Offset: 1

Author

John M. Campbell, May 12 2012

Keywords

• sign
• tabl
• hard
• nice

Comments

The determinant of the n X n such matrix has a closed form given in the formula section (and the Mathematica code below).

Rows appear to be given by polynomials (see formula section).

Examples

			Evaluate the fifth derivative of sin(sin(sin(x))) at 0, which is 33. So the (3,3) entry of the array is 33. The array begins as:
|  1      1        1         1         1          1 |
| -1     -2       -3        -4        -5         -6 |
|  1     12       33        64       105        156 |
| -1   -128     -731     -2160     -4765      -8896 |
|  1   1872    25857    121600    368145     873936 |
| -1 -37600 -1311379 -10138880 -42807605 -130426016 |
		

Crossrefs

Cf. A003712, A003715.

Programs

Maple

A:= (i, j)-> (D@@(2*i-1))(sin@@j)(0):
seq(seq(A(i, 1+d-i), i=1..d), d=1..9); # Alois P. Heinz, May 14 2012

Mathematica

A[a_, b_] :=
  A[a, b] =
   Array[D[Nest[Sin, x, #2], {x, 2*#1 - 1}] /. x -> 0 &, {a, b}];
Print[A[7, 7] // MatrixForm];
Table2 = {};
k = 1;
While[k < 8, Table1 = {};
  i = 1;
  j = k;
  While[0 < j,
   AppendTo[Table1, First[Take[First[Take[A[7, 7], {i, i}]], {j, j}]]];
   j = j - 1;
   i = i + 1];
  AppendTo[Table2, Table1];
  k++];
Print[Flatten[Table2]]
Print[Table[Det[A[n, n]], {n, 1, 7}]];
Print[Table[(
  I^(n + n^2) 2^(-(1/12) + n^2) 3^(n/2 - n^2/2)
    Glaisher^3 (-(1/\[Pi]))^
   n BarnesG[1/2 + n] BarnesG[1 + n] BarnesG[3/2 + n])/E^(1/4), {n, 1, 7}]]

Formula

A(i,j) = ((d/dx)^(2i-1) sin^j(x))_{x=0}.

Let A_n denote the n X n such matrix. Then:

det(A_n)=(i^(n + n^2) 2^(-(1/12) + n^2) 3^(n/2 - n^2/2) G^3 (-(1/pi))^n B(1/2 + n) B(1 + n) B(3/2 + n))/e^(1/4), where B is the Barnes G-function and G is the Glaisher-Kinkelin constant (and i is the imaginary unit). (This can be shown by evaluating recurrence relations for det(A_n)). See Mathematica code below.

First row: 1.

Second row: -x.

Third row: x (5 x - 4).

Fourth row: -(1/3) x (164 + 7 x (-48 + 25 x)).

Fifth row: (8 - 7 x)^2 x (-24 + 25 x).

Sixth row: -(1/3) x (213568 - 766656 x + 1004696 x^2 - 572880 x^3 + 121275 x^4).

Seventh row: 1/3 x (-14371328 + 65012064 x - 111160192 x^2 + 91291200 x^3 - 36552516 x^4 + 5780775 x^5).

Second column: A003712.

Third column: A003715.