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 A091917 #22 Jan 22 2020 05:19:22
%S A091917 1,-2,1,0,-2,1,-2,3,-3,1,0,-4,6,-4,1,-2,5,-10,10,-5,1,0,-6,15,-20,15,
%T A091917 -6,1,-2,7,-21,35,-35,21,-7,1,0,-8,28,-56,70,-56,28,-8,1,-2,9,-36,84,
%U A091917 -126,126,-84,36,-9,1,0,-10,45,-120,210,-252,210,-120,45,-10,1,-2,11,-55,165,-330,462,-462,330,-165,55,-11,1
%N A091917 Coefficient array of polynomials (z-1)^n-1.
%C A091917 The first element has been changed to 1 to produce an invertible matrix. Alternatively, this is the coefficient array for the polynomials P(z,n) = Product_{j=0..n-1} (z-(1+w(n)^j)) where w(n) = e^(2*Pi*i/n), i=sqrt(-1).
%C A091917 The row entries determine interesting recurrences. For instance, a(n) = 4a(n-1) + 6a(n-2) + 4a(n-3), a(0)=a(1)=a(2)=1, gives A038503. Sequences of the form a(n) = Sum_{k=0..n} (binomial(n,k) if k mod m = r, otherwise 0), for r=0..m-1, result. Equivalently, a(n) = Sum_{j=0..n-1} 2^n*(cos(Pi*j/m))^n*cos((n-2r)Pi*j/m)/m, r=0..m-1. These include A024493, A024494, A024495, A038503, A038504, A038505. The inverse matrix is A091918.
%C A091917 Triangle T(n,k), 0 <= k <= n, read by rows given by [ -2, 2, 1/2, -1/2, 0, 0, 0, 0, 0, ...] DELTA [1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Oct 11 2007
%H A091917 Alois P. Heinz, <a href="/A091917/b091917.txt">Rows n = 0..140, flattened</a>
%F A091917 T(n,k) = T(n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,1) = T(2,2) = 1, T(1,0) = T(2,1) = -2, T(2,0) = 0, T(n,k) = 0 for k > n or for k < 0. - _Philippe Deléham_, May 23 2015
%F A091917 G.f.: (1-2*x-x^2+x^2*y)/((x-1)*(-x+x*y-1)). - _R. J. Mathar_, Aug 11 2015
%e A091917 Rows begin:
%e A091917 { 1},
%e A091917 {-2, 1},
%e A091917 { 0, -2, 1},
%e A091917 {-2, 3, -3, 1},
%e A091917 { 0, -4, 6, -4, 1},
%e A091917 ...
%p A091917 T:= n-> `if`(n=0, 1, (p-> seq(coeff(p,z,i), i=0..n))((z-1)^n-1)):
%p A091917 seq(T(n), n=0..12); # _Alois P. Heinz_, May 23 2015
%t A091917 Table[If[n == 0, 1, CoefficientList[(z-1)^n-1, z]], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Apr 08 2016 *)
%o A091917 (PARI) row(n) = if (n==0, 1, Vecrev((z-1)^n-1)); \\ _Michel Marcus_, May 23 2015
%K A091917 sign,tabl
%O A091917 0,2
%A A091917 _Paul Barry_, Feb 13 2004