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 A347147 #45 Aug 16 2024 20:48:59
%S A347147 1,1,1,1,2,1,1,4,4,1,1,7,10,7,1,1,12,23,23,12,1,1,20,50,62,50,20,1,1,
%T A347147 33,104,156,156,104,33,1,1,54,211,373,438,373,211,54,1,1,88,420,859,
%U A347147 1155,1155,859,420,88,1,1,143,824,1925,2915,3306,2915,1925,824,143,1
%N A347147 Square array read by antidiagonals: T(n,k) is the number of rook paths from (1,1) to (n,k) if the rook may travel 1 to i squares along rank or file i, n >= 1, k >= 1.
%C A347147 Note that all of the rook moves are in the positive horizontal or vertical direction.
%C A347147 By symmetry, the array is equal to its transpose.
%C A347147 From the definition, T(1,1) = 1 and T(n,k) = Sum_{i=n-k..n-1} T(i,k) + Sum_{j=k-n..k-1} T(n,j) if we take T(n,k)=0 for n<=0 or k<=0.
%H A347147 Alissa S. Crans and Glen T. Whitney, <a href="https://bookstore.ams.org/PRB/38">The Mathematical Playground: People and Problems from 31 Years of Math Horizons</a>, AMS/MAA Problem Books (2024) Vol. 38. Problem 419, pp. 112-113.
%F A347147 T(n,k) = 2*(T(n-1,k)+T(n,k-1))-3T(n-1,k-1)-T(n,k-n-1)+T(n-1,k-n), for 1<n<k (can be shown by inclusion-exclusion on the paths); and symmetrically for 1<k<n.
%F A347147 T(n,n) = 2*(T(n-1,n)+T(n,n-1))-3T(n-1,n-1) = 4T(n-1,n)-3T(n-1,n-1), for n>1.
%e A347147 There are four rook paths with move length capped by the number of the rank or file it is moving along, from (1,1) to (3,2):
%e A347147 (1,1)->(2,1)->(3,1)->(3,2);
%e A347147 (1,1)->(2,1)->(2,2)->(3,2);
%e A347147 (1,1)->(1,2)->(2,2)->(3,2);
%e A347147 (1,1)->(1,2)->(3,2).
%e A347147 So T(3,2) = 4.
%e A347147 An initial portion of the full array:
%e A347147 n= 1 2 3 4 5 6 7 8 9 ...
%e A347147 -----------------------------------------
%e A347147 k=1: 1 1 1 1 1 1 1 1 1 ...
%e A347147 k=2: 1 2 4 7 12 20 33 54 88 ...
%e A347147 k=3: 1 4 10 23 50 104 211 420 824 ...
%e A347147 k=4: 1 7 23 62 156 373 859 1925 4226 ...
%e A347147 k=5: 1 12 50 156 438 1155 2915 7114 16917 ...
%e A347147 k=6: 1 20 104 373 1155 3306 8978 23450 59422 ...
%e A347147 ....
%o A347147 (Python)
%o A347147 n = 1; k = 1;
%o A347147 T = [[],[0]] # Dummy 0th entry, and dummy [1][0]th entry.
%o A347147 T[n].append(1) # set T[1][1] to 1
%o A347147 print(f"T(1,1) = {T[n][k]}")
%o A347147 for m in range(64):
%o A347147 if n == 1:
%o A347147 n = k + 1; k = 1;
%o A347147 T.append([0]); # initialize T[n], with dummy 0th entry.
%o A347147 else:
%o A347147 n -= 1; k += 1;
%o A347147 T[n].append(sum(T[i][k] for i in range(max(1,n-k),n))
%o A347147 + sum(T[n][j] for j in range(max(1,k-n),k)))
%o A347147 print(f"T({n},{k}) = {T[n][k]}")
%Y A347147 Cf. A000071 (row n=2, and column k=2).
%Y A347147 Cf. A035002 (unlimited rook moves).
%Y A347147 A347148 gives a similar array that includes the 0 file and rank.
%K A347147 nonn,tabl
%O A347147 1,5
%A A347147 _Glen Whitney_, Aug 20 2021