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).
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
KEYWORD
AUTHOR
John M. Campbell, May 12 2012
STATUS
approved