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 begins: 1; 1,2; 3,4,4; 19,24,16,8; 243,304,192,64,16; 6227,7776,4864,1536,256,32; 319251,398528,248832,77824,12288,1024,64; 32737427,40864128,25505792,7962624,1245184,98304,4096,128; ... Matrix square, T^2, equals SHIFT_UP(T): 1; 3,4; 19,24,16; 243,304,192,64; 6227,7776,4864,1536,256; 319251,398528,248832,77824,12288,1024; ... G.f. for column 0: 1 = (1-x) + 1*x*(1-x)(1-2x) + 3*x^2*(1-x)(1-2x)(1-4x) + ... + T(n,0)*x^n*(1-x)(1-2x)(1-4x)*..*(1-2^n*x) + ... G.f. for column 1: 2 = 2(1-2x) + 4*x*(1-2x)(1-4x) + 24*x^2*(1-2x)(1-4x)(1-8x) + ... + T(n+1,1)*x^n*(1-2x)(1-4x)(1-8x)*..*(1-2^(n+1)*x) + ... G.f. for column 2: 4 = 4(1-4x) + 16*x*(1-4x)(1-8x) + 192*x^2*(1-4x)(1-8x)(1-16x) + ... + T(n+2,2)*x^n*(1-4x)(1-8x)(1-16x)*..*(1-2^(n+2)*x) + ...
{T(n, k)=local(A=matrix(1, 1), B); A[1, 1]=1; for(m=2, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i, i]=2^(i-1), if(j==1, B[i, j]=(A^2)[i-1, 1], B[i, j]=(A^2)[i-1, j])); )); A=B); return(A[n+1, k+1])}
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