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 A162169 #45 Sep 18 2021 17:44:59
%S A162169 1,-1,1,-1,0,1,1,0,-3,1,1,0,-6,0,1,-1,0,10,0,-5,1,-1,0,15,0,-15,0,1,1,
%T A162169 0,-21,0,35,0,-7,1,1,0,-28,0,70,0,-28,0,1,-1,0,36,0,-126,0,84,0,-9,1,
%U A162169 -1,0,45,0,-210,0,210,0,-45,0,1,1,0,-55,0,330,0,-462,0,165,0,-11,1
%N A162169 Exponential series expansion of (cos(x) - sin(x))*cosh(t*x) + sinh(t*x).
%C A162169 Previous name was: Signed version of Pascal's triangle.
%C A162169 Related to A000111 via its matrix inverse A162170.
%C A162169 For odd columns k, T(n, k) = binomial(n-1, k-1) * (-1)^floor((n+k-1)/2). For even columns, T(n, k) = 1 if n = k, otherwise 0. - _Mike Tryczak_, Jun 17 2015
%C A162169 From _Peter Bala_, Sep 08 2021: (Start)
%C A162169 In the notation of the Bala link, this is the array [[ cos(x) - sin(x), 1 ]] with inverse array A162170 = [[ sec(x) + tan(x), 1 ]].
%C A162169 In general, arrays of the form [[ G(x), 1 ]], where G(x) = 1 + g(1)*x + g(2)*x^2/2! + g(3)*x^3/3! + ... is an e.g.f., form a group with group law [[ G(x), 1 ]]*[[ F(x), 1 ]] = [[ G(x)*F_e(x) + F_o(x), 1 ]] and inverse array [[ G(x), 1 ]]^(-1) = [[ (1 - G_o(x))/G_e(x), 1 ]], where G_e(x) = (G(x) + G(-x))/2 and G_o(x) = (G(x) - G(-x))/2 are the even and odd parts of G(x). (End)
%H A162169 Peter Bala, <a href="/A177994/a177994.pdf">Matrices with repeated columns - the generalised Appell groups</a>
%F A162169 E.g.f.: (cos(x) - sin(x))*cosh(t*x) + sinh(t*x) = 1 + (-1 + t)*x + (-1 + t^2)*x^2/2! + (1 - 3^t^2 + t^3)*x^3/3! + .... - _Peter Bala_, Sep 08 2021
%e A162169 Table begins:
%e A162169 1;
%e A162169 -1, 1;
%e A162169 -1, 0, 1;
%e A162169 1, 0, -3, 1;
%e A162169 1, 0, -6, 0, 1;
%e A162169 -1, 0, 10, 0, -5, 1;
%e A162169 -1, 0, 15, 0, -15, 0, 1;
%e A162169 1, 0, -21, 0, 35, 0, -7, 1;
%e A162169 1, 0, -28, 0, 70, 0, -28, 0, 1;
%e A162169 -1, 0, 36, 0, -126, 0, 84, 0, -9, 1;
%e A162169 -1, 0, 45, 0, -210, 0, 210, 0, -45, 0, 1;
%e A162169 1, 0, -55, 0, 330, 0, -462, 0, 165, 0, -11, 1;
%e A162169 .
%e A162169 As a symmetric triangle:
%e A162169 1;
%e A162169 -1, 1;
%e A162169 -1, 0, 1;
%e A162169 1, 0, -3, 1;
%e A162169 1, 0, -6, 0, 1;
%e A162169 -1, 0, 10, 0, -5, 1;
%e A162169 -1, 0, 15, 0, -15, 0, 1;
%e A162169 1, 0, -21, 0, 35, 0, -7, 1;
%e A162169 1, 0, -28, 0, 70, 0, -28, 0, 1;
%e A162169 -1, 0, 36, 0, -126, 0, 84, 0, -9, 1;
%e A162169 -1, 0, 45, 0, -210, 0, 210, 0, -45, 0, 1;
%e A162169 1, 0, -55, 0, 330, 0, -462, 0, 165, 0, -11, 1;
%p A162169 egf := (cos(x) - sin(x))*cosh(t*x) + sinh(t*x):
%p A162169 ser := n -> series(egf, x, n+1): c := n -> n!*coeff(ser(n), x, n):
%p A162169 A162169row := n -> seq(coeff(c(n), t, k), k=0..n):
%p A162169 for n from 0 to 9 do A162169row(n) od; # _Peter Luschny_, Sep 18 2021
%t A162169 nn=12; Flatten[Table[Table[If[Or[Mod[n - k, 4] == 1, Mod[n - k, 4] == 2], -1, 1]*If[n >= k, Binomial[n - 1, k - 1], 0]*If[And[n > k, Mod[k, 2] == 0], 0, 1], {k, 1, n}], {n, 1, nn}]] (* _Mats Granvik_, Nov 25 2017 *)
%o A162169 (Excel) =if(or(mod(row()-column();4)=1;mod(row()-column();4)=2);-1;1)*if(row()>=column();combin(row()-1;column()-1);0)*if(and(row()>column();mod(column();2)=0);0;1)
%o A162169 (PARI) T(n, k) = if (k % 2, binomial(n-1, k-1) * (-1)^floor((n+k-1)/2), if (n==k, 1 , 0));
%o A162169 tabl(nn) = {for (n=1, nn, for (k=1, n, print1(T(n,k), ", ");); print(););} \\ _Michel Marcus_, Jun 17 2015
%Y A162169 Cf. A007318, A162170.
%K A162169 sign,tabl
%O A162169 1,9
%A A162169 _Mats Granvik_, Jun 27 2009
%E A162169 New name using a formula of _Peter Bala_ from _Peter Luschny_, Sep 18 2021