This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A147308 #34 Mar 28 2020 06:37:32
%S A147308 1,0,1,-1,0,1,0,-4,0,1,5,0,-10,0,1,0,40,0,-20,0,1,-61,0,175,0,-35,0,1,
%T A147308 0,-768,0,560,0,-56,0,1,1385,0,-4996,0,1470,0,-84,0,1,0,24320,0,
%U A147308 -22720,0,3360,0,-120,0,1,-50521,0,214445,0
%N A147308 Riordan array [sech(x), arcsin(tanh(x))].
%C A147308 Production array is [cos(x),x] beheaded. Inverse is A147309. Row sums are A012123(n+1).
%C A147308 If signs are ignored this is identical to A147309. - _N. J. A. Sloane_, Nov 07 2008
%C A147308 The Bell transform of the Euler numbers. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 18 2016
%H A147308 Nathaniel Johnston, <a href="/A147308/b147308.txt">Table of n, a(n) for n = 0..229</a>
%F A147308 From _Vladimir Kruchinin_, Dec 18 2011: (Start)
%F A147308 [gd(x)]^m = sum(n>=m T(n,m)*m!/n!*x^n), where gd(x) is the Gudermannian function.
%F A147308 T(n,m) = sum(j=0..(n-m)/2, (sum(i=0..2*j, (2^(i)*Stirling1(i+m,m) *C(2*j+m-1,i+m-1))/(i+m)!)) *sum(k=0..n-2*j-m, (-1)^(k+j) *C(k+2*j+m-1,2*j+m-1) *(k+2*j+m)! *2^(-k-2*j) *Stirling2(n,k+2*j+m))), n>=m>=1.
%F A147308 (End)
%e A147308 Triangle begins:
%e A147308 1;
%e A147308 0, 1;
%e A147308 -1, 0, 1;
%e A147308 0, -4, 0, 1;
%e A147308 5, 0, -10, 0, 1;
%e A147308 0, 40, 0, -20, 0, 1;
%e A147308 -61, 0, 175, 0, -35, 0, 1;
%p A147308 Z := proc(n,x) option remember; if n = 0 then return 1: else return 1/2*x*(Z(n-1, x-1)+Z(n-1, x+1)): fi:end:
%p A147308 with(PolynomialTools): for n from 1 to 10 do for k from 1 to n do printf("%d, ",(-1)^floor((n-k)/2)*coeff(Z(n,x), x, k)):od: printf("\n"):od: # _Nathaniel Johnston_, Apr 21 2011
%t A147308 t[n_, k_] := SeriesCoefficient[ 2^k*ArcTan[(E^x - 1)/(E^x + 1)]^k*n!/k!, {x, 0, n}]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jan 23 2015 *)
%t A147308 Z[n_, x_] := Z[n, x] = If[n == 0, 1, x*(Z[n-1, x-1] + Z[n-1, x+1])/2 // Simplify]; t[n_, k_] := (-1)^Floor[(n-k)/2]*Coefficient[Z[n, x], x, k]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Feb 27 2015, after Maple *)
%o A147308 (Sage) # uses[bell_matrix from A264428]
%o A147308 # Adds a column 1,0,0,0, ... at the left side of the triangle.
%o A147308 bell_matrix(euler_number, 10) # _Peter Luschny_, Jan 18 2016
%K A147308 easy,sign,tabl
%O A147308 0,8
%A A147308 _Paul Barry_, Nov 05 2008