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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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%I #23 Dec 18 2015 18:18:11

%S 1,1,2,1,4,16,1,6,72,272,1,8,168,2896,7936,1,10,304,10672,203904,

%T 353792,1,12,480,26400,1198080,22112000,22368256,1,14,696,52880,

%U 4071040,208521728,3412366336,1903757312,1,16,952,92912,10373760,976629760,51874413568,709998153728,209865342976

%N 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.

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

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NestedFunction.html">Nested Function</a>

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

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

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

%e Array A(i,j) begins:

%e . 1, 1, 1, 1, 1, ...

%e . 2, 4, 6, 8, 10, ...

%e . 16, 72, 168, 304, 480, ...

%e . 272, 2896, 10672, 26400, 52880, ...

%e . 7936, 203904, 1198080, 4071040, 10373760, ...

%e . 353792, 22112000, 208521728, 976629760, 3172514560, ...

%e Evaluate the (2*3-1)th derivate of tan(tan(tan(x))) at 0, which is 168. Thus A(3,3)=168.

%p A:= (i, j)-> (D@@(2*i-1))(tan@@j)(0):

%p seq(seq(A(i, 1+d-i), i=1..d), d=1..8); # _Alois P. Heinz_, May 13 2012

%t A[a_, b_] :=

%t A[a, b] =

%t Array[D[Nest[Tan, x, #2], {x, 2*#1 - 1}] /. x -> 0 &, {a, b}];

%t Print[A[7, 7] // MatrixForm];

%t Table2 = {};

%t k = 1;

%t While[k < 8, Table1 = {};

%t i = 1;

%t j = k;

%t While[0 < j,

%t AppendTo[Table1, First[Take[First[Take[A[7, 7], {i, i}]], {j, j}]]];

%t j = j - 1;

%t i = i + 1];

%t AppendTo[Table2, Table1];

%t k++];

%t Print[Flatten[Table2]];

%t Print[Table[Det[A[n, n]], {n, 1, 7}]];

%t Table[(2^(11/12 +

%t 1/2 (5 + 3 (-1 + n)) (-1 + n)) 3^(-(1/2) (-1 +

%t n) n) Glaisher^3 \[Pi]^-n BarnesG[1/2 + n] BarnesG[1 + n] BarnesG[3/2 + n])/E^(1/4), {n, 1, 7}]

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

%K nonn,tabl,hard

%O 1,3

%A _John M. Campbell_, May 12 2012

%E More terms from _Alois P. Heinz_, May 13 2012