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.
Triangle T(n,k) begins: 1; 0, 1; 0, 0, 1; 0, 1, 0, 1; 0, 0, 2, 0, 1; 0, 0, 0, 3, 0, 1; 0, 0, 2, 0, 4, 0, 1; 0, 0, 0, 5, 0, 5, 0, 1; 0, 0, 0, 0, 9, 0, 6, 0, 1; 0, 0, 0, 5, 0, 14, 0, 7, 0, 1; 0, 0, 0, 0, 14, 0, 20, 0, 8, 0, 1; 0, 0, 0, 0, 0, 28, 0, 27, 0, 9, 0, 1; 0, 0, 0, 0, 14, 0, 48, 0, 35, 0, 10, 0, 1; ...
A165408:= func< n,k | n le 3*k select Binomial(Floor((n+k)/2), k)*((3*k-n)/2 +1)*(1+(-1)^(n-k))/(2*(k+1)) else 0 >; [A165408(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Nov 09 2022
b:= proc(x, y) option remember; `if`(y<=x, `if`(x=0, 1,
b(x-1, y)+`if`(y>0, b(x, y-1), 0)), 0)
end:
T:= (n, k)-> `if`((n-k)::even, b(k, (n-k)/2), 0):
seq(seq(T(n, k), k=0..n), n=0..14); # Alois P. Heinz, Sep 20 2022
b[x_, y_]:= b[x, y]= If[y<=x, If[x==0, 1, b[x-1, y] + If[y>0, b[x, y-1], 0]], 0];
T[n_, k_]:= If[EvenQ[n-k], b[k, (n-k)/2], 0];
Table[T[n, k], {n,0,14}, {k,0,n}]//Flatten (* Jean-François Alcover, Oct 08 2022, after Alois P. Heinz *)
def A165408(n,k): return 0 if (n>3*k) else binomial(int((n+k)/2), k)*((3*k-n+2)/2 )*(1+(-1)^(n-k))/(2*(k+1)) flatten([[A165408(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Nov 09 2022
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