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 A080233 #22 Mar 25 2025 11:59:42
%S A080233 1,1,0,1,1,-1,1,2,0,-2,1,3,2,-2,-3,1,4,5,0,-5,-4,1,5,9,5,-5,-9,-5,1,6,
%T A080233 14,14,0,-14,-14,-6,1,7,20,28,14,-14,-28,-20,-7,1,8,27,48,42,0,-42,
%U A080233 -48,-27,-8,1,9,35,75,90,42,-42,-90,-75,-35,-9
%N A080233 Triangle T(n,k) obtained by taking differences of consecutive pairs of row elements of Pascal's triangle A007318.
%C A080233 Row sums are 1,1,1,1,1,1 with g.f. 1/(1-x). Can also be obtained from triangle A080232 by taking sums of pairs of consecutive row elements.
%C A080233 Mirror image of triangle in A156644. - _Philippe Deléham_, Feb 14 2009
%F A080233 T(n, k) = if(k>n, 0, binomial(n, k)-binomial(n, k-1)).
%e A080233 Triangle begins as:
%e A080233 1;
%e A080233 1, 0;
%e A080233 1, 1, -1;
%e A080233 1, 2, 0, -2;
%e A080233 1, 3, 2, -2, -3;
%e A080233 1, 4, 5, 0, -5, -4;
%e A080233 1, 5, 9, 5, -5, -9, -5;
%e A080233 1, 6, 14, 14, 0, -14, -14, -6;
%e A080233 1, 7, 20, 28, 14, -14, -28, -20, -7;
%e A080233 1, 8, 27, 48, 42, 0, -42, -48, -27, -8;
%e A080233 1, 9, 35, 75, 90, 42, -42, -90, -75, -35, -9;
%e A080233 ...
%t A080233 Table[Binomial[n, k] - Binomial[n, k - 1], {n, 0, 9}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Nov 24 2016 *)
%o A080233 (PARI) {T(n, k) = if( n<0 || k>n, 0, binomial(n, k) - binomial(n, k-1))}; /* _Michael Somos_, Nov 25 2016 */
%Y A080233 Row sums give A000012.
%Y A080233 Cf. A007318, A080232.
%K A080233 easy,sign,tabl
%O A080233 0,8
%A A080233 _Paul Barry_, Feb 10 2003