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 A171840 #21 Feb 13 2022 09:28:29
%S A171840 1,1,2,1,2,5,1,2,4,15,1,2,4,9,52,1,2,4,8,23,203,1,2,4,8,17,65,877,1,2,
%T A171840 4,8,16,40,199,4140,1,2,4,8,16,33,104,654,21147,1,2,4,8,16,32,73,291,
%U A171840 2296,115975,1,2,4,8,16,32,65,177,857,8569,678570
%N A171840 Triangle read by rows, truncated columns of an array formed by taking sets of P(n) = Pascal's triangle, with the 1's column shifted up n = 1,2,3,... times. Then the n-th row of the array = lim_{k->infinity}, k=1,2,3,...; (P(n))^k, deleting the first 1.
%C A171840 Row sums = A171841: (1, 3, 8, 22, 68, 241, 974, ...).
%C A171840 Right border = the Bell sequence A000110 starting (1, 2, 5, 15, 52, ...).
%C A171840 Row 2 of the array = A007476 starting (1, 1, 2, 4, 9, 23, 65, 199, ...).
%F A171840 Triangle read by rows, truncated columns of an array formed by taking sets of P(n) = Pascal's triangle, with the 1's column shifted up n = 1,2,3,... times. Then n-th row of the array = lim_{k->infinity} (P(n))^k, deleting the first 1.
%e A171840 First few rows of the array:
%e A171840 1, 2, 5, 15, 52, 203, 877, 4140, 21147, ...
%e A171840 1, 1, 2, 4, 9, 23, 65, 199, 654, ...
%e A171840 1, 1, 1, 2, 4, 8, 17, 40, 104, ...
%e A171840 1, 1, 1, 1, 2, 4, 8, 16, 33, ...
%e A171840 1, 1, 1, 1, 1, 2, 4, 8, 16, ...
%e A171840 ...
%e A171840 Rightmost diagonal of 1's becomes leftmost column of the triangle:
%e A171840 1;
%e A171840 1, 2;
%e A171840 1, 2, 5;
%e A171840 1, 2, 4, 15;
%e A171840 1, 2, 4, 9, 52;
%e A171840 1, 2, 4, 8, 23, 203;
%e A171840 1, 2, 4, 8, 17, 65, 877;
%e A171840 1, 2, 4, 8, 16, 40, 199, 4140;
%e A171840 1, 2, 4, 8, 16, 33, 104, 654, 21147;
%e A171840 1, 2, 4, 8, 16, 32, 73, 291, 2296, 115975;
%e A171840 1, 2, 4, 8, 16, 32, 65, 177, 857, 8569, 678570;
%e A171840 ...
%e A171840 Example: n-th row corresponds to P(n) = Pascal's triangle with 1's column shifted up 1 row, so that P(1) =
%e A171840 1;
%e A171840 1;
%e A171840 1, 1;
%e A171840 1, 2, 1;
%e A171840 1, 3, 3, 1;
%e A171840 ...
%e A171840 then take lim_{k->infinity} (P(1))^k, getting A000110: (1, 1, 2, 5, 15, 52, ...), then delete the first 1.
%o A171840 (Sage)
%o A171840 # generates the diagonals of the triangle, starting with diag = 1 the Bell numbers.
%o A171840 def A171840_generator(len, diag) :
%o A171840 A = [1]*diag
%o A171840 for n in (0..len) :
%o A171840 for k in range(n, 0, -1) :
%o A171840 A[k - 1] += A[k]
%o A171840 A.append(A[0])
%o A171840 yield A[0]
%o A171840 for diag in (1..5) : print(list(A171840_generator(10, diag)))
%o A171840 # _Peter Luschny_, Feb 27 2012
%Y A171840 Cf. A007318, A007476, A171841, A000110.
%K A171840 nonn,tabl
%O A171840 1,3
%A A171840 _Gary W. Adamson_, Dec 19 2009