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 A230206 #44 Sep 08 2022 08:46:06
%S A230206 -1,3,-3,1,-1,2,0,-2,1,-1,1,2,-2,-1,1,-1,0,3,0,-3,0,1,-1,-1,3,3,-3,-3,
%T A230206 1,1,-1,-2,2,6,0,-6,-2,2,1,-1,-3,0,8,6,-6,-8,0,3,1,-1,-4,-3,8,14,0,
%U A230206 -14,-8,3,4,1,-1,-5,-7,5,22,14,-14,-22,-5,7
%N A230206 Trapezoid of dot products of row 3 (signs alternating) with sequential 4-tuples read by rows in Pascal's triangle A007318: T(n,k) is the linear combination of the 4-tuples (C(3,0), -C(3,1), C(3,2), -C(3,3)) and (C(n-1,k-3), C(n-1,k-2), C(n-1,k-1), C(n-1,k)), n >= 1, 0 <= k <= n+2.
%C A230206 The array is trapezoidal rather than triangular because C(n,k) is not uniquely defined for all negative n and negative k.
%C A230206 Row sums are 0.
%C A230206 Coefficients of (x-1)^3 (x+1)^(n-1) for n > 0.
%H A230206 Dixon J. Jones, <a href="/A230206/b230206.txt">Rows n = 1..100 for irregular triangle, flattened</a>
%H A230206 Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, <a href="https://www.emis.de/journals/JIS/VOL21/Falcao/falcao2.html">Combinatorial Identities Associated with a Multidimensional Polynomial Sequence</a>, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
%F A230206 T(n,k) = Sum_{i=0..n+m-1} (-1)^(i+m)*C(m,i)*C(n-1,k-i), n >= 1, with T(n,0) = (-1)^m and m=3.
%e A230206 Trapezoid begins
%e A230206 -1, 3, -3, 1;
%e A230206 -1, 2, 0, -2, 1;
%e A230206 -1, 1, 2, -2, -1, 1;
%e A230206 -1, 0, 3, 0, -3, 0, 1;
%e A230206 -1, -1, 3, 3, -3, -3, 1, 1;
%e A230206 -1, -2, 2, 6, 0, -6, -2, 2, 1;
%e A230206 -1, -3, 0, 8, 6, -6, -8, 0, 3, 1;
%t A230206 Flatten[Table[CoefficientList[(x - 1)^3 (x + 1)^n, x], {n, 0, 7}]] (* _T. D. Noe_, Oct 25 2013 *)
%t A230206 m=3; Table[If[k == 0, (-1)^m, Sum[(-1)^(j+m)*Binomial[m, j]*Binomial[n-1, k-j], {j, 0, n+m-1}]], {n, 1, 10}, {k, 0, n+m-1}]//Flatten (* _G. C. Greubel_, Nov 29 2018 *)
%o A230206 (PARI) m=3; for(n=1, 10, for(k=0, n+m-1, print1(if(k==0, (-1)^m, sum(j=0, n+m-1, (-1)^(j+m)*binomial(m,j)*binomial(n-1,k-j))), ", "))) \\ _G. C. Greubel_, Nov 29 2018
%o A230206 (Magma) m:=3; [[k le 0 select (-1 )^m else (&+[(-1)^(j+m)* Binomial(m,j) *Binomial(n-1,k-j): j in [0..(n+m-1)]]): k in [0..(n+m-1)]]: n in [1..10]]; // _G. C. Greubel_, Nov 29 2018
%o A230206 (Sage) m=3; [[sum((-1)^(j+m)*binomial(m,j)*binomial(n-1,k-j) for j in range(n+m)) for k in range(n+m)] for n in (1..10)] # _G. C. Greubel_, Nov 29 2018
%Y A230206 Using row j of the alternating Pascal triangle as generator: A007318 (j=0), A008482 and A112467 (j=1 after the first term in each), A182533 (j=2 after the first two rows), A230207-A230212 (j=4 to j=9).
%K A230206 easy,sign,tabf
%O A230206 1,2
%A A230206 _Dixon J. Jones_, Oct 11 2013