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 A363882 #33 Aug 02 2023 14:07:40 %S A363882 0,1,1,0,2,0,1,3,3,1,0,4,4,4,0,1,5,10,10,5,1,0,6,12,20,12,6,0,1,7,21, %T A363882 35,35,21,7,1,0,8,24,56,64,56,24,8,0,1,9,36,84,126,126,84,36,9,1,0,10, %U A363882 40,120,200,252,200,120,40,10,0,1,11,55,165,330,462,462,330,165,55,11,1 %N A363882 Take 2 copies of Pascal's triangle. One copy has one inch between the terms of each row and the other copy has two inches between the terms of each row. Put one on top of the other so that the 1's at the very top of each copy coincide. Sequence is a triangle giving the differences between the overlapping terms. %F A363882 T(n,k) = binomial(n,k) - [n mod 2 = k mod 2 = 0] * binomial(n/2,k/2). %e A363882 Row n=8 is formed by taking row 8 of Pascal's triangle (1, 8, 28, 56, 70, 56, 28, 8, 1) and subtracting row 4 (1, 4, 6, 4, 1) spaced 2 apart. The numbers that overlap are 1, 28, 70, 28, 1 over 1, 4, 6, 4, 1, from which 1-1=0, 28-4=24, 70-6=64, 28-4=24, and 1-1=0. Thus, row 8 of the present triangle is 0, 8, 24, 56, 64, 56, 24, 8, 0. %e A363882 Subtracting: %e A363882 1; 1; 0, %e A363882 1, 1; 1, 1; %e A363882 1, 2, 1; - 1, 1; = 0, 2, 0; %e A363882 1, 3, 3, 1; 1, 3, 3, 1; %e A363882 1, 4, 6, 4, 1; 1, 2, 1; 0, 4, 4, 4, 0; %e A363882 ... %e A363882 Resulting triangle begins: %e A363882 k=0 1 2 3 4 %e A363882 n=0: 0; %e A363882 n=1: 1, 1; %e A363882 n=2: 0, 2, 0; %e A363882 n=3: 1, 3, 3, 1; %e A363882 n=4: 0, 4, 4, 4, 0; %e A363882 ... %Y A363882 Cf. A007318. %K A363882 tabl,nonn,easy %O A363882 0,5 %A A363882 _J. Lowell_, Jun 25 2023