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 A093966 #14 Dec 30 2021 23:26:06
%S A093966 1,1,2,1,4,3,1,6,9,4,1,6,21,16,5,1,6,33,52,25,6,1,6,33,124,105,36,7,1,
%T A093966 6,33,196,345,186,49,8,1,6,33,196,825,786,301,64,9,1,6,33,196,1305,
%U A093966 2586,1561,456,81,10,1,6,33,196,1305,6186,6601,2808,657,100,11
%N A093966 Array read by antidiagonals: number of {112,221}-avoiding words.
%C A093966 A(n,k) is the number of n-long k-ary words that simultaneously avoid the patterns 112 and 221.
%H A093966 G. C. Greubel, <a href="/A093966/b093966.txt">Antidiagonals n = 1..50, flattened</a>
%H A093966 A. Burstein and T. Mansour, <a href="https://arxiv.org/abs/math/0110056">Words restricted by patterns with at most 2 distinct letters</a>, arXiv:math/0110056 [math.CO], 2001.
%F A093966 A(n, k) = k!*binomial(n, k) + Sum_{j=1..k-1} j*j!*binomial(n, j), for 2 <= k <= n, otherwise Sum_{j=1..n} j*j!*binomial(n, j), with A(1, k) = 1 and A(n, 1) = n.
%F A093966 From _G. C. Greubel_, Dec 29 2021: (Start)
%F A093966 T(n, k) = A(k, n-k+1).
%F A093966 Sum_{k=1..n} T(n, k) = A093963(n).
%F A093966 T(n, 1) = 1.
%F A093966 T(n, n) = n.
%F A093966 T(n, n-1) = (n-1)^2.
%F A093966 T(n, n-2) = A069778(n).
%F A093966 T(2*n-1, n) = A093965(n).
%F A093966 T(2*n, n) = A093964(n), for n >= 1. (End)
%e A093966 Array, A(n, k), begins as:
%e A093966 1, 1, 1, 1, 1, 1, 1 ... 1*A000012(k);
%e A093966 2, 4, 6, 6, 6, 6, 6 ... 2*A158799(k-1);
%e A093966 3, 9, 21, 33, 33, 33, 33 ... ;
%e A093966 4, 16, 52, 124, 196, 196, 196 ... ;
%e A093966 5, 25, 105, 345, 825, 1305, 1305 ... ;
%e A093966 6, 36, 186, 786, 2586, 6186, 9786 ... ;
%e A093966 7, 49, 301, 1561, 6601, 21721, 51961 ... ;
%e A093966 Antidiagonal triangle, T(n, k), begins as:
%e A093966 1;
%e A093966 1, 2;
%e A093966 1, 4, 3;
%e A093966 1, 6, 9, 4;
%e A093966 1, 6, 21, 16, 5;
%e A093966 1, 6, 33, 52, 25, 6;
%e A093966 1, 6, 33, 124, 105, 36, 7;
%e A093966 1, 6, 33, 196, 345, 186, 49, 8;
%e A093966 1, 6, 33, 196, 825, 786, 301, 64, 9;
%e A093966 1, 6, 33, 196, 1305, 2586, 1561, 456, 81, 10;
%t A093966 A[n_, k_]:= A[n, k]= If[n==1, 1, If[k==1, n, If[2<=k<n+1, (1-k)*k!*Binomial[n, k] + Sum[j*j!*Binomial[n, j], {j, k}], Sum[j*j!*Binomial[n, j], {j, n}] ]]];
%t A093966 T[n_, k_]:= A[k, n-k+1];
%t A093966 Table[T[k, k], {n, 15}, {k, n}]//Flatten (* _G. C. Greubel_, Dec 29 2021 *)
%o A093966 (PARI) A(n,k) = if(n >= k+1, sum(j=1, k, j*j!*binomial(k,j)), if(n<2, if(n<1, 0, k), n!*binomial(k,n) + sum(j=1, n-1, j*j!*binomial(k,j))));
%o A093966 T(n,k) = A(n-k+1, k);
%o A093966 for(n=1, 15, for(k=1, n, print1(T(n, k), ", ") ) )
%o A093966 (Sage)
%o A093966 @CachedFunction
%o A093966 def A(n,k):
%o A093966 if (n==1): return 1
%o A093966 elif (k==1): return n
%o A093966 elif (2 <= k < n+1): return factorial(k)*binomial(n,k) + sum( j*factorial(j)*binomial(n,j) for j in (1..k-1) )
%o A093966 else: return sum( j*factorial(j)*binomial(n,j) for j in (1..n) )
%o A093966 def T(n,k): return A(k, n-k+1)
%o A093966 flatten([[T(n,k) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, Dec 29 2021
%Y A093966 Cf. A069778, A093963 (antidiagonal sums), A093964, A093965 (main diagonal).
%K A093966 nonn,tabl
%O A093966 1,3
%A A093966 _Ralf Stephan_, Apr 20 2004