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

1,5

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).

Links

Table of n, a(n) for n=1..45.

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.

Example

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 |

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}]]

Crossrefs

Cf. A003712, A003715.

Sequence in context: A220898 A336163 A066013 * A014521 A084389 A122022

Adjacent sequences: A212258 A212259 A212260 * A212262 A212263 A212264

Keyword

sign,tabl,hard,nice

Author

John M. Campbell, May 12 2012

Status

approved