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

%I A166318 #7 Jun 28 2018 04:43:32
%S A166318 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,
%T A166318 -140,0,1,0,-49152,0,8960,0,-224,0,1,354560,0,-319744,0,23520,0,-336,
%U A166318 0,1,0,6225920,0,-1454080,0,53760,0,-480,0,1,-51733504,0,54897920,0
%N A166318 Exponential Riordan array [sech(2x), arctan(tanh(x))].
%C A166318 Inverse is A166317. Row sums are A012222(n+1). Signed version of A166317.
%C A166318 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
%e A166318 Triangle begins
%e A166318 1,
%e A166318 0, 1,
%e A166318 -4, 0, 1,
%e A166318 0, -16, 0, 1,
%e A166318 80, 0, -40, 0, 1,
%e A166318 0, 640, 0, -80, 0, 1,
%e A166318 -3904, 0, 2800, 0, -140, 0, 1,
%e A166318 0, -49152, 0, 8960, 0, -224, 0, 1,
%e A166318 354560, 0, -319744, 0, 23520, 0, -336, 0, 1,
%e A166318 0, 6225920, 0, -1454080, 0, 53760, 0, -480, 0, 1,
%e A166318 Production matrix is
%e A166318 0, 1,
%e A166318 -4, 0, 1,
%e A166318 0, -12, 0, 1,
%e A166318 16, 0, -24, 0, 1,
%e A166318 0, 80, 0, -40, 0, 1,
%e A166318 -64, 0, 240, 0, -60, 0, 1,
%e A166318 0, -448, 0, 560, 0, -84, 0, 1,
%e A166318 256, 0, -1792, 0, 1120, 0, -112, 0, 1,
%e A166318 0, 2304, 0, -5376, 0, 2016, 0, -144, 0, 1,
%e A166318 -1024, 0, 11520, 0, -13440, 0, 3360, 0, -180, 0, 1
%e A166318 which is the exponential Riordan array [cos(2x),x] minus its top row.
%p A166318 # The function BellMatrix is defined in A264428.
%p A166318 # Adds (1,0,0,0, ..) as column 0.
%p A166318 BellMatrix(n -> 2^n*euler(n), 10); # _Peter Luschny_, Jan 29 2016
%t A166318 BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A166318 B = BellMatrix[Function[n, 2^n EulerE[n]], rows = 12];
%t A166318 Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)
%K A166318 easy,sign,tabl
%O A166318 0,4
%A A166318 _Paul Barry_, Oct 11 2009