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 begins: S(1) = (2) / (2^2 * 0!), S(2) = -(8 - Pi^2) / (2^4 * 1!) = A123092, S(3) = (64 - 6*Pi^2) / (2^6 * 2!) = A248895, S(4) = -(768 - 60*Pi^2 - 2*Pi^4)/ (2^8 * 3!) = A248896, S(5) = (12288 - 840*Pi^2 - 40*Pi^4) / (2^10 * 4!), S(6) = -(245760 - 15120*Pi^2 - 840*Pi^4 - 16*Pi^6) / (2^12 * 5!), S(7) = (5898240 - 332640*Pi^2 - 20160*Pi^4 - 672*Pi^6) / (2^14 * 6!), S(8) = -(165150720 - 8648640*Pi^2 - 554400*Pi^4 - 24192*Pi^6 - 272*Pi^8) / (2^16 * 7!), S(9) = (5284823040 - 259459200*Pi^2 - 17297280*Pi^4 - 887040*Pi^6 - 19584*Pi^8) / (2^18 * 8!), ...
T(2,1) = 3 because there are 3 length 4 binary sequences in which the third zero appears in the fourth position: {0,0,1,0}, {0,1,0,0}, {1,0,0,0}.
Triangle begins
1;
1, 1;
3, 3, 2;
10, 10, 8, 4;
35, 35, 30, 20, 8;
126, 126, 112, 84, 48, 16;
...
T:= (n, k)-> ceil(binomial(2*n-k, n)*2^(k-1)): seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Apr 06 2025
Table[Table[Binomial[2 n - k, n]*2^(k - 1), {k, 0, n}], {n, 0, 9}] // Grid
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