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 A347148 #19 May 04 2023 01:57:21
%S A347148 1,1,1,1,2,1,1,3,3,1,1,4,8,4,1,1,5,16,16,5,1,1,6,30,42,30,6,1,1,7,53,
%T A347148 98,98,53,7,1,1,8,91,216,268,216,91,8,1,1,9,153,455,677,677,455,153,9,
%U A347148 1,1,10,254,931,1627,1906,1627,931,254,10,1
%N A347148 Square array read by antidiagonals: T(n,0) = T(0,k) = 1 and for n > 0, k > 0, T(n,k) = Sum_{i=1..min(n,k)} (T(n-i,k) + T(n,k-i)).
%C A347148 From the definition, this is the number of ways a rook can get to square (n,k) if it is allowed to enter at any square (n,0) or (0,k), and can proceed 1 to i squares to the right along rank i or up file i at each move thereafter.
%C A347148 By symmetry, the array is equal to its transpose.
%C A347148 In its triangular form produced by "rotating the array an eighth turn", it is generated by a recurrence that can be stated as "each entry is the sum of the largest symmetric V of entries that converge at the given location" (see the triangle in Examples).
%F A347148 T(n,k) = 2*(T(n-1,k) + T(n,k-1)) - 3*T(n-1,k-1) - T(n,k-n-1) + T(n-1,k-n), for 1 < n < k; and symmetrically for 1 < k < n; identical to the formula for A347147.
%F A347148 T(n,n) = 2*(T(n-1,n) + T(n,n-1)) - 3*T(n-1,n-1) + 2 = 4*T(n-1,n) - 3*T(n-1,n-1) + 2, for n > 1.
%e A347148 Initial values of T:
%e A347148 T(1,1) = T(1,0) + T(0,1) = 2,
%e A347148 T(2,1) = T(1,1) + T(2,0) = 3 = T(1,2),
%e A347148 T(3,1) = T(2,1) + T(3,0) = 4,
%e A347148 T(2,2) = T(1,2) + T(2,1) + T(0,2) + T(2,0) = 8,
%e A347148 T(3,2) = T(2,2) + T(3,1) + T(1,2) + T(3,0) = 16.
%e A347148 An initial portion of the full array:
%e A347148 n= 0 1 2 3 4 5 6 7 8 ...
%e A347148 --------------------------------------
%e A347148 k=0: 1 1 1 1 1 1 1 1 1 ...
%e A347148 k=1: 1 2 3 4 5 6 7 8 9 ...
%e A347148 k=2: 1 3 8 16 30 53 91 153 254 ...
%e A347148 k=3: 1 4 16 42 98 216 455 931 1866 ...
%e A347148 k=4: 1 5 30 98 268 677 1627 3763 8465 ...
%e A347148 k=5: 1 6 53 216 677 1906 5039 12747 31180 ...
%e A347148 ....
%e A347148 As a triangle: the _underlined_ entries add up to the *starred* one, making a symmetric "V", the largest possible at that position:
%e A347148 1
%e A347148 1 1
%e A347148 1 2 1
%e A347148 1 3 3 1
%e A347148 _1_ 4 _8_ 4 1
%e A347148 1 _5_ _16_ 16 5 1
%e A347148 1 6 *30* 42 30 6 1
%e A347148 ......
%o A347148 (Python)
%o A347148 T = [[1,1],[1],[1]] # set T[0][0]=T[1][0]=T[0][1]=T[0,2]=1
%o A347148 print(f"T(0, 0) = {T[0][0]}")
%o A347148 print(f"T(1, 0) = {T[1][0]}")
%o A347148 print(f"T(0, 1) = {T[0][1]}")
%o A347148 print(f"T(2, 0) = {T[2][0]}")
%o A347148 n=2; k=0;
%o A347148 for j in range(54):
%o A347148 if n == 1:
%o A347148 T[0].append(1); # set T[0][k+1] to 1
%o A347148 print(f"T({0}, {k+1}) = {T[0][k+1]}")
%o A347148 T.append([1]); # set T[k+2][0] to 1
%o A347148 n = k+2; k = 0;
%o A347148 print(f"T({n}, 0) = {T[n][0]}")
%o A347148 continue;
%o A347148 n -= 1; k += 1;
%o A347148 T[n].append(sum(T[n-i][k]+T[n][k-i] for i in range(1,min(n,k)+1)))
%o A347148 print(f"T({n}, {k}) = {T[n][k]}")
%Y A347148 Cf. A347147 (in which the rook can only enter at (1,1), but moves identically).
%K A347148 nonn,tabl
%O A347148 1,5
%A A347148 _Glen Whitney_, Aug 21 2021