A212267 Array A(i,j) read by antidiagonals: A(i,j) is the (2*i-1)-th derivative of tan(tan(tan(...tan(x)))) nested j times evaluated at 0.

1, 1, 2, 1, 4, 16, 1, 6, 72, 272, 1, 8, 168, 2896, 7936, 1, 10, 304, 10672, 203904, 353792, 1, 12, 480, 26400, 1198080, 22112000, 22368256, 1, 14, 696, 52880, 4071040, 208521728, 3412366336, 1903757312, 1, 16, 952, 92912, 10373760, 976629760, 51874413568, 709998153728, 209865342976

Offset

1,3

Comments

The determinant of the n X n such matrix has a closed form given in the Mathematica code below.

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

Links

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

Eric Weisstein's World of Mathematics, Nested Function

Formula

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

Third row: n*(5*n - 1)*4 = 8*A005476(n).

Fourth row: 8/3*n*(11 - 84*n + 175*n^2).

Example

Array A(i,j) begins:
.      1,        1,         1,         1,          1, ...
.      2,        4,         6,         8,         10, ...
.     16,       72,       168,       304,        480, ...
.    272,     2896,     10672,     26400,      52880, ...
.   7936,   203904,   1198080,   4071040,   10373760, ...
. 353792, 22112000, 208521728, 976629760, 3172514560, ...
Evaluate the (2*3-1)th derivate of tan(tan(tan(x))) at 0, which is 168. Thus A(3,3)=168.

Maple

A:= (i, j)-> (D@@(2*i-1))(tan@@j)(0):
seq(seq(A(i, 1+d-i), i=1..d), d=1..8); # Alois P. Heinz, May 13 2012

Mathematica

A[a_, b_] :=
  A[a, b] =
   Array[D[Nest[Tan, 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}]];
Table[(2^(11/12 +
       1/2 (5 + 3 (-1 + n)) (-1 + n)) 3^(-(1/2) (-1 +
         n) n) Glaisher^3 \[Pi]^-n BarnesG[1/2 + n] BarnesG[1 + n] BarnesG[3/2 + n])/E^(1/4), {n, 1, 7}]

Crossrefs

Columns j=1-3 give: A000182, A003718, A003720.

Sequence in context: A263575 A162977 A032174 * A344110 A346793 A087801

Adjacent sequences: A212264 A212265 A212266 * A212268 A212269 A212270

Keyword

nonn,tabl,hard

Author

John M. Campbell, May 12 2012

Extensions

More terms from Alois P. Heinz, May 13 2012

Status

approved