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.
{a(n)=local(M=matrix(n+3,n+3,r,c,binomial(2^(c-1),r-c)));(M^-1)[n+3,3]}
{a(n)=local(M=matrix(n+1,n+1,r,c,binomial(2^(c-1),r-c))); sum(k=0,n,(M^-1)[n+1,k+1])}
Triangle begins: 1; 1, 1; 0, 2, 1; 0, 1, 4, 1; 0, 0, 6, 8, 1; 0, 0, 4, 28, 16, 1; 0, 0, 1, 56, 120, 32, 1; 0, 0, 0, 70, 560, 496, 64, 1; 0, 0, 0, 56, 1820, 4960, 2016, 128, 1; 0, 0, 0, 28, 4368, 35960, 41664, 8128, 256, 1; 0, 0, 0, 8, 8008, 201376, 635376, 341376, 32640, 512, 1; 0, 0, 0, 1, 11440, 906192, 7624512, 10668000, 2763520, 130816, 1024, 1;
[Binomial(2^k, n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 15 2021
Table[Binomial[2^k, n-k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 15 2021 *)
T(n,k)=binomial(2^k,n-k)
flatten([[binomial(2^k, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 15 2021