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 A368175 #33 Dec 21 2023 02:54:28
%S A368175 1,1,1,1,2,2,1,2,3,3,1,2,4,6,6,1,2,4,7,10,10,1,2,4,8,14,20,20,1,2,4,8,
%T A368175 15,25,35,35,1,2,4,8,16,30,50,70,70,1,2,4,8,16,31,56,91,126,126,1,2,4,
%U A368175 8,16,32,62,112,182,252,252,1,2,4,8,16,32,63,119,210,336,462,462
%N A368175 Square array read by ascending antidiagonals: T(n,k) = Sum_{i=ceiling((k-n)/2)..floor((k+n-1)/2)} binomial(k,i), with n >= 1, k >= 0.
%C A368175 T(n,k), for k >= 1, is the size of the largest possible set S of k-bit strings such that, if S_a < S_b are members of S, then W(S_b) < W(S_a) + n, where W is A000120.
%C A368175 T(1,k), for k >= 1, gives the number of rows in the Christmas tree pattern (cf. A367508) of order k. Furthermore, T(n,k), for k >= 1, gives the number of rows generated by iteratively applying k times the map described in A367508, starting from a single row of length n.
%D A368175 Donald E. Knuth, The Art of Computer Programming, Vol. 4A: Combinatorial Algorithms, Part 1, Addison-Wesley, 2011, Section 7.2.1.6, exercises 71 and 72, pp. 479 and 799.
%H A368175 Paolo Xausa, <a href="/A368175/b368175.txt">Table of n, a(n) for n = 1..11325</a> (antidiagonals 1..150 of the array, flattened).
%F A368175 T(n,0) = 1.
%F A368175 T(1,k) = A001405(k).
%F A368175 T(n,k) = 2^k = A000079(k), for n > k.
%F A368175 T(n,n) = 2^n - 1 = A000225(n).
%F A368175 Antidiagonal sums: Sum_{n=1..d} T(n,d-n) = binomial(d+1,floor((d+1)/2)) - 1 = A014495(d+1), for d >= 1.
%e A368175 Array begins:
%e A368175 n\k| 0 1 2 3 4 5 6 7 8 9 10 ...
%e A368175 ---+--------------------------------------------------
%e A368175 1 | 1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, ... = A001405
%e A368175 2 | 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 462, ... = A001405
%e A368175 3 | 1, 2, 4, 7, 14, 25, 50, 91, 182, 336, 672, ... = A026010
%e A368175 4 | 1, 2, 4, 8, 15, 30, 56, 112, 210, 420, 792, ... = A026023
%e A368175 5 | 1, 2, 4, 8, 16, 31, 62, 119, 238, 456, 912, ...
%e A368175 6 | 1, 2, 4, 8, 16, 32, 63, 126, 246, 492, 957, ...
%e A368175 7 | 1, 2, 4, 8, 16, 32, 64, 127, 254, 501, 1002, ...
%e A368175 8 | 1, 2, 4, 8, 16, 32, 64, 128, 255, 510, 1012, ...
%e A368175 9 | 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1022, ...
%e A368175 10 | 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, ...
%e A368175 ...
%e A368175 For n = 3 and k = 4 the 14 members of S are 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110.
%t A368175 A368175[n_,k_]:=If[n>k,2^k,Sum[Binomial[k,i],{i,Ceiling[(k-n)/2],Floor[(k+n-1)/2]}]];
%t A368175 With[{dmax=15},Table[A368175[n-k,k],{n,dmax},{k,0,n-1}]] (* Generates 15 antidiagonals *)
%Y A368175 Cf. A000079, A000120, A000225, A001405, A014495, A026010, A026023, A191781, A367508.
%K A368175 nonn,tabl,easy
%O A368175 1,5
%A A368175 _Paolo Xausa_, Dec 14 2023