A166318 Exponential Riordan array [sech(2x), arctan(tanh(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

Offset: 0

Author

Paul Barry, Oct 11 2009

Keywords

• easy
• sign
• tabl

Comments

Inverse is A166317. Row sums are A012222(n+1). Signed version of A166317.

Also the Bell transform of the sequence a(n) = 2^n*E(n) (E(n) the Euler numbers) without column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 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,
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,
-1024, 0, 11520, 0, -13440, 0, 3360, 0, -180, 0, 1
which is the exponential Riordan array [cos(2x),x] minus its top row.
		

Programs

Maple

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> 2^n*euler(n), 10); # Peter Luschny, Jan 29 2016

Mathematica

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, 2^n EulerE[n]], rows = 12];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)