A166317 Exponential Riordan array [sec(2x), arctanh(tan(x))].

1, 0, 1, 4, 0, 1, 0, 16, 0, 1, 80, 0, 40, 0, 1, 0, 640, 0, 80, 0, 1, 3904, 0, 2800, 0, 140, 0, 1, 0, 49152, 0, 8960, 0, 224, 0, 1, 354560, 0, 319744, 0, 23520, 0, 336, 0, 1, 0, 6225920, 0, 1454080, 0, 53760, 0, 480, 0, 1, 51733504, 0, 54897920, 0, 5230720, 0, 110880, 0, 660, 0, 1

Offset: 0

Author

Paul Barry, Oct 11 2009

Keywords

• easy
• nonn
• tabl

Comments

The Bell transform of abs(2^n*euler_number(n)). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

Examples

			Triangle begins
  1;
  0, 1;
  4, 0, 1;
  0, 16, 0, 1;
  80, 0, 40, 0, 1;
  0, 640, 0, 80, 0, 1;
  3904, 0, 2800, 0, 140, 0, 1;
  0, 49152, 0, 8960, 0, 224, 0, 1;
  354560, 0, 319744, 0, 23520, 0, 336, 0, 1;
  0, 6225920, 0, 1454080, 0, 53760, 0, 480, 0, 1;
  51733504, 0, 54897920, 0, 5230720, 0, 110880, 0, 660, 0, 1;
Production matrix is
    0,    1;
    4,    0,    1;
    0,   12,    0,    1;
   16,    0,   24,    0,    1;
    0,   80,    0,   40,    0,    1;
   64,    0,  240,    0,   60,    0,   1;
    0,  448,    0,  560,    0,   84,   0,   1;
  256,    0, 1792,    0, 1120,    0, 112,   0, 1;
    0, 2304,    0, 5376,    0, 2016,   0, 144, 0, 1;
which is the exponential Riordan array [cosh(2x),x] minus its top row. (Cf. also A117435.)
		

Crossrefs

Row sums are A012259(n+1).

Inverse is A166318 which is a signed version of this sequence.

Cf. A117435, A264428.

Programs

Mathematica

(* The function BellMatrix is defined in A264428. *)
rows = 12;
M = BellMatrix[Abs[2^#*EulerE[#]]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jul 11 2019 *)

Sage

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: abs(2^n*euler_number(n)), 10) # Peter Luschny, Jan 18 2016