A003718 -id:A003718 - cp's OEIS frontend

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

John M. Campbell, May 12 2012

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).

			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.

Eric Weisstein's World of Mathematics, Nested Function

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

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

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}]

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).

More terms from Alois P. Heinz, May 13 2012

A302453 a(n) = coefficient of x^(2*n-1) in the n-th iteration (n-fold self-composition) of e.g.f. tan(x).

Original entry on oeis.org

1, 4, 168, 26400, 10373760, 8226518272, 11524607732736, 26047611675267072, 88935060882222120960, 436394080487109570265088, 2959343413232671759344861184, 26874522377891724867898947141632, 318464577992023576681854032513335296, 4818779071094868918454887699722367139840

Offset: 1

Ilya Gutkovskiy, Apr 08 2018

nonn

a(n) = coefficient of x^(2*n-1) in the n-th iteration (n-fold self-composition) of e.g.f. tanh(x) (absolute values).

			The initial coefficients of successive iterations of e.g.f. A(x) = tan(x) (odd powers only) are as follows:
n = 1: (1),  2,   16,    272,      7936,  ... e.g.f. A(x)
n = 2:  1,  (4),  72,   2896,    203904,  ... e.g.f. A(A(x))
n = 3:  1,   6, (168), 10672,   1198080,  ... e.g.f. A(A(A(x)))
n = 4:  1,   8,  304, (26400),  4071040,  ... e.g.f. A(A(A(A(x))))
n = 5:  1,  10,  480,  52880, (10373760), ... e.g.f. A(A(A(A(A(x)))))
...
More explicitly, the successive iterations of e.g.f. A(x) = tan(x) begin:
tan(x) = x/1! + 2*x^3/3! + 16*x^5/5! + 272*x^7/7! + 7936*x^9/9! + ...
tan(tan(x)) = x/1! + 4*x^3/3! + 72*x^5/5! + 2896*x^7/7! + 203904*x^9/9! + ...
tan(tan(tan(x))) = x/1! + 6*x^3/3! + 168*x^5/5! + 10672*x^7/7! + 1198080*x^9/9! + ...
tan(tan(tan(tan(x)))) = x/1! + 8*x^3/3! + 304*x^5/5! + 26400*x^7/7! + 4071040*x^9/9! + ...
tan(tan(tan(tan(tan(x))))) = x/1! + 10*x^3/3! + 480*x^5/5! + 52880*x^7/7! + 10373760*x^9/9! + ...

N. J. A. Sloane, Transforms

Cf. A000182, A003718, A003720.

Table[(2 n - 1)! SeriesCoefficient[Nest[Function[x, Tan[x]], x, n], {x, 0, 2 n - 1}], {n, 14}]

cp's OEIS Frontend

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.

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