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 A230211 #29 Sep 08 2022 08:46:06
%S A230211 1,-8,28,-56,70,-56,28,-8,1,1,-7,20,-28,14,14,-28,20,-7,1,1,-6,13,-8,
%T A230211 -14,28,-14,-8,13,-6,1,1,-5,7,5,-22,14,14,-22,5,7,-5,1,1,-4,2,12,-17,
%U A230211 -8,28,-8,-17,12,2,-4,1,1,-3,-2,14,-5,-25,20,20,-25,-5,14
%N A230211 Trapezoid of dot products of row 8 (signs alternating) with sequential 9-tuples read by rows in Pascal's triangle A007318: T(n,k) is the linear combination of the 9-tuples (C(8,0), -C(8,1), ..., -C(8,7), C(8,8)) and (C(n-1,k-8), C(n-1,k-7), ..., C(n-1,k)), n >= 1, 0 <= k <= n+7.
%C A230211 The array is trapezoidal rather than triangular because C(n,k) is not uniquely defined for all negative n and negative k.
%C A230211 Row sums are 0.
%C A230211 Coefficients of ((x-1)^8)(x+1)^(n-1), n > 0.
%H A230211 G. C. Greubel, <a href="/A230211/b230211.txt">Rows n=1..50 of trapezoid, flattened</a>
%H A230211 Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, 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 A230211 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=8.
%e A230211 Trapezoid begins:
%e A230211 1, -8, 28, -56, 70, -56, 28, -8, 1;
%e A230211 1, -7, 20, -28, 14, 14, -28, 20, -7, 1;
%e A230211 1, -6, 13, -8, -14, 28, -14, -8, 13, -6, 1;
%e A230211 1, -5, 7, 5, -22, 14, 14, -22, 5, 7, -5, 1;
%e A230211 1, -4, 2, 12, -17, -8, 28, -8, -17, 12, 2, -4, 1;
%e A230211 1, -3, -2, 14, -5, -25, 20, 20, -25, -5, 14, -2, -3, 1;
%e A230211 1, -2, -5, 12, 9, -30, -5, 40, -5, -30, 9, 12, -5, -2, 1;
%e A230211 etc.
%t A230211 Flatten[Table[CoefficientList[(x - 1)^8 (x + 1)^n, x], {n, 0, 7}]] (* _T. D. Noe_, Oct 25 2013 *)
%t A230211 m=8; 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 28 2018 *)
%o A230211 (PARI) m=8; 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 28 2018
%o A230211 (Magma) m:=8; [[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 28 2018
%o A230211 (Sage) m=8; [[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 28 2018
%Y A230211 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), A230206-A230210 (j=3 to j=7), A230212 (j=9).
%K A230211 easy,sign,tabf
%O A230211 1,2
%A A230211 _Dixon J. Jones_, Oct 12 2013