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 A096470 #25 Aug 09 2020 01:07:55
%S A096470 1,1,1,1,0,1,1,-1,2,1,1,-2,4,-3,1,1,-3,7,-10,11,1,1,-4,11,-21,32,-31,
%T A096470 1,1,-5,16,-37,69,-100,101,1,1,-6,22,-59,128,-228,329,-328,1,1,-7,29,
%U A096470 -88,216,-444,773,-1101,1102,1,1,-8,37,-125,341,-785,1558,-2659,3761,-3760,1,1,-9,46,-171,512,-1297,2855,-5514,9275,-13035,13036,1
%N A096470 Triangle T(n,k), read by rows, formed by setting all entries in the zeroth column and in the main diagonal ((n,n) entries) to 1 and defining the rest of the entries by the recursion T(n,k) = T(n-1,k) - T(n,k-1).
%C A096470 If A(x,y) is the bivariate o.g.f. of a triangular array T(n,k) and B(x,y) is the bivariate o.g.f. of its mirror image T(n,n-k), then B(x,y) = A(x*y, y^(-1)) and A(x,y) = B(x*y, y^(-1)). - _Petros Hadjicostas_, Aug 08 2020
%F A096470 T(n,k) = T(n-1,k) - T(n,k-1) for 1 <= k <= n-1 with T(n,0) = 1 = T(n,n) for n >= 0.
%F A096470 The 2nd column is T(n,2) = A000124(n-2) for n >= 2 (Hogben's central polygonal numbers).
%F A096470 The "first subdiagonal" (unsigned) is |T(n,n-1)| = A032357(n-1) for n >= 1 (Convolution of Catalan numbers and powers of -1).
%F A096470 The "2nd subdiagonal" (unsigned) is |T(n,n-2)| = A033297(n) = Sum_{i=0..n-2} (-1)^i*C(n-1-i) for n >= 2, where C(n) are the Catalan numbers (A000108).
%F A096470 From _Petros Hadjicostas_, Aug 08 2020: (Start)
%F A096470 |T(n,k)| = |A168377(n,n-k)| for 0 <= k <= n.
%F A096470 Bivariate o.g.f.: (1 + y + x*y*c(-x*y))/((1 - x*y)*(1 - x + y)), where c(x) = 2/(1 + sqrt(1 - 4*x)) = o.g.f. of A000108.
%F A096470 Bivariate o.g.f. of |T(n,k)|: (1 - y - x*y*c(x*y))/((1 + x*y)*(1 - x - y)) + 2*x*y/(1 - x^2*y^2).
%F A096470 Bivariate o.g.f. of mirror image T(n,n-k): (1 + y + x*y*c(-x))/((1 - x)*(1 + y - x*y^2)).
%F A096470 Bivariate o.g.f. of |T(n,n-k)|: (1 - y + x*y*c(x))/((1 + x)*(1 - y + x*y^2)) + 2*x/(1 - x^2). (End)
%e A096470 From _Petros Hadjicostas_, Aug 08 2020: (Start)
%e A096470 Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
%e A096470 1;
%e A096470 1, 1;
%e A096470 1, 0, 1;
%e A096470 1, -1, 2, 1;
%e A096470 1, -2, 4, -3, 1;
%e A096470 1, -3, 7, -10, 11, 1;
%e A096470 1, -4, 11, -21, 32, -31, 1;
%e A096470 1, -5, 16, -37, 69, -100, 101, 1;
%e A096470 1, -6, 22, -59, 128, -228, 329, -328, 1;
%e A096470 ... (End)
%o A096470 (PARI) T(n, k) = if ((k==0) || (n==k), 1, if ((n<0) || (k<0), 0, if (n>k, T(n-1, k) - T(n, k-1), 0)));
%o A096470 for(n=0, 10, for (k=0, n, print1(T(n, k), ", ")); print); \\ _Petros Hadjicostas_, Aug 08 2020
%Y A096470 Cf. A000108, A000124, A032357, A033297, A168377.
%K A096470 sign,tabl
%O A096470 0,9
%A A096470 _Gerald McGarvey_, Aug 12 2004
%E A096470 Offset changed to 0 by _Petros Hadjicostas_, Aug 08 2020