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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. 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 (list; table; graph; refs; listen; history; text; internal format)
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 * A087801 A238454 A025228

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

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Last modified November 13 04:20 EST 2019. Contains 329085 sequences. (Running on oeis4.)