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 A047072 #26 Oct 13 2022 04:54:52
%S A047072 1,1,1,1,2,1,1,1,1,1,1,2,2,2,1,1,3,2,2,3,1,1,4,5,4,5,4,1,1,5,9,5,5,9,
%T A047072 5,1,1,6,14,14,10,14,14,6,1,1,7,20,28,14,14,28,20,7,1,1,8,27,48,42,28,
%U A047072 42,48,27,8,1,1,9,35,75,90,42,42,90,75,35,9,1
%N A047072 Array A read by diagonals: A(h,k)=number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right and no step touches the line y=x unless x=0 or x=h.
%H A047072 G. C. Greubel, <a href="/A047072/b047072.txt">Antidiagonals n = 0..50, flattened</a>
%H A047072 R. K. Guy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
%F A047072 A(n, n) = 2*[n=0] - A002420(n),
%F A047072 A(n, n+1) = 2*A000108(n-1), n >= 1.
%F A047072 From _G. C. Greubel_, Oct 13 2022: (Start)
%F A047072 T(n, n-1) = A000027(n-2) + 2*[n<3], n >= 1.
%F A047072 T(n, n-2) = A000096(n-4) + 2*[n<5], n >= 2.
%F A047072 T(n, n-3) = A005586(n-6) + 4*[n<7] - 2*[n=3], n >= 3.
%F A047072 T(2*n, n) = 2*A000108(n-1) + 3*[n=0].
%F A047072 T(2*n-1, n-1) = T(2*n+1, n+1) = A000180(n).
%F A047072 T(3*n, n) = A025174(n) + [n=0]
%F A047072 Sum_{k=0..n} T(n, k) = 2*A063886(n-2) + [n=0] - 2*[n=1]
%F A047072 Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n).
%F A047072 Sum_{k=0..floor(n/2)} T(n, k) = A047079(n). (End)
%e A047072 Array, A(n, k), begins as:
%e A047072 1, 1, 1, 1, 1, 1, 1, 1, ...;
%e A047072 1, 2, 1, 2, 3, 4, 5, 6, ...;
%e A047072 1, 1, 2, 2, 5, 9, 14, 20, ...;
%e A047072 1, 2, 2, 4, 5, 14, 28, 48, ...;
%e A047072 1, 3, 5, 5, 10, 14, 42, 90, ...;
%e A047072 1, 4, 9, 14, 14, 28, 42, 132, ...;
%e A047072 1, 5, 14, 28, 42, 42, 84, 132, ...;
%e A047072 1, 6, 20, 48, 90, 132, 132, 264, ...;
%e A047072 Antidiagonals, T(n, k), begins as:
%e A047072 1;
%e A047072 1, 1;
%e A047072 1, 2, 1;
%e A047072 1, 1, 1, 1;
%e A047072 1, 2, 2, 2, 1;
%e A047072 1, 3, 2, 2, 3, 1;
%e A047072 1, 4, 5, 4, 5, 4, 1;
%e A047072 1, 5, 9, 5, 5, 9, 5, 1;
%e A047072 1, 6, 14, 14, 10, 14, 14, 6, 1;
%t A047072 A[_, 0]= 1; A[0, _]= 1; A[h_, k_]:= A[h, k]= If[(k-1>h || k-1<h) && h != k-1, A[h, k-1], 0] + If[h-1 != k, A[h-1, k], 0];
%t A047072 Table[A[h-k, k], {h,0,11}, {k,h,0, -1}]//Flatten (* _Jean-François Alcover_, Mar 06 2019 *)
%o A047072 (Magma)
%o A047072 b:= func< n | n eq 0 select 1 else 2*Catalan(n-1) >;
%o A047072 function A(n,k)
%o A047072 if k eq n then return b(n);
%o A047072 elif k gt n then return Binomial(n+k-1, n) - Binomial(n+k-1, n-1);
%o A047072 else return Binomial(n+k-1, k) - Binomial(n+k-1, k-1);
%o A047072 end if; return A;
%o A047072 end function;
%o A047072 // [[A(n,k): k in [0..12]]: n in [0..12]];
%o A047072 T:= func< n,k | A(n-k, k) >;
%o A047072 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 13 2022
%o A047072 (SageMath)
%o A047072 def A(n,k):
%o A047072 if (k==n): return 2*catalan_number(n-1) + 2*int(n==0)
%o A047072 elif (k>n): return binomial(n+k-1, n) - binomial(n+k-1, n-1)
%o A047072 else: return binomial(n+k-1, k) - binomial(n+k-1, k-1)
%o A047072 def T(n,k): return A(n-k, k)
%o A047072 # [[A(n,k) for k in range(12)] for n in range(12)]
%o A047072 flatten([[T(n,k) for k in range(n+1)] for n in range(12)]) # _G. C. Greubel_, Oct 13 2022
%Y A047072 Cf. A000007, A000027, A000096, A002420.
%Y A047072 Cf. A005586, A025174, A047079, A063886.
%Y A047072 Cf. A047073, A047074, A047079.
%Y A047072 The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
%Y A047072 Diagonals give A000108, A000245, A002057, A000344, A003517, A000588, A003518, A003519, A001392, ...
%K A047072 nonn,tabl
%O A047072 0,5
%A A047072 _Clark Kimberling_