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

%I A166317 #25 Mar 28 2020 14:02:36
%S A166317 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,
%T A166317 1,0,49152,0,8960,0,224,0,1,354560,0,319744,0,23520,0,336,0,1,0,
%U A166317 6225920,0,1454080,0,53760,0,480,0,1,51733504,0,54897920,0,5230720,0,110880,0,660,0,1
%N A166317 Exponential Riordan array [sec(2x), arctanh(tan(x))].
%C A166317 The Bell transform of abs(2^n*euler_number(n)). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 18 2016
%e A166317 Triangle begins
%e A166317   1;
%e A166317   0, 1;
%e A166317   4, 0, 1;
%e A166317   0, 16, 0, 1;
%e A166317   80, 0, 40, 0, 1;
%e A166317   0, 640, 0, 80, 0, 1;
%e A166317   3904, 0, 2800, 0, 140, 0, 1;
%e A166317   0, 49152, 0, 8960, 0, 224, 0, 1;
%e A166317   354560, 0, 319744, 0, 23520, 0, 336, 0, 1;
%e A166317   0, 6225920, 0, 1454080, 0, 53760, 0, 480, 0, 1;
%e A166317   51733504, 0, 54897920, 0, 5230720, 0, 110880, 0, 660, 0, 1;
%e A166317 Production matrix is
%e A166317     0,    1;
%e A166317     4,    0,    1;
%e A166317     0,   12,    0,    1;
%e A166317    16,    0,   24,    0,    1;
%e A166317     0,   80,    0,   40,    0,    1;
%e A166317    64,    0,  240,    0,   60,    0,   1;
%e A166317     0,  448,    0,  560,    0,   84,   0,   1;
%e A166317   256,    0, 1792,    0, 1120,    0, 112,   0, 1;
%e A166317     0, 2304,    0, 5376,    0, 2016,   0, 144, 0, 1;
%e A166317 which is the exponential Riordan array [cosh(2x),x] minus its top row. (Cf. also A117435.)
%t A166317 (* The function BellMatrix is defined in A264428. *)
%t A166317 rows = 12;
%t A166317 M = BellMatrix[Abs[2^#*EulerE[#]]&, rows];
%t A166317 Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jul 11 2019 *)
%o A166317 (Sage) # uses[bell_matrix from A264428]
%o A166317 # Adds a column 1,0,0,0, ... at the left side of the triangle.
%o A166317 bell_matrix(lambda n: abs(2^n*euler_number(n)), 10) # _Peter Luschny_, Jan 18 2016
%Y A166317 Row sums are A012259(n+1).
%Y A166317 Inverse is A166318 which is a signed version of this sequence.
%Y A166317 Cf. A117435, A264428.
%K A166317 easy,nonn,tabl
%O A166317 0,4
%A A166317 _Paul Barry_, Oct 11 2009