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.
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, 4, 8, 16, 40, 199, 4140, 1, 2, 4, 8, 16, 33, 104, 654, 21147, 1, 2, 4, 8, 16, 32, 73, 291, 2296, 115975, 1, 2, 4, 8, 16, 32, 65, 177, 857, 8569, 678570
Offset: 1
Examples
First few rows of the array:
1, 2, 5, 15, 52, 203, 877, 4140, 21147, ...
1, 1, 2, 4, 9, 23, 65, 199, 654, ...
1, 1, 1, 2, 4, 8, 17, 40, 104, ...
1, 1, 1, 1, 2, 4, 8, 16, 33, ...
1, 1, 1, 1, 1, 2, 4, 8, 16, ...
...
Rightmost diagonal of 1's becomes leftmost column of the triangle:
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, 4, 8, 16, 40, 199, 4140;
1, 2, 4, 8, 16, 33, 104, 654, 21147;
1, 2, 4, 8, 16, 32, 73, 291, 2296, 115975;
1, 2, 4, 8, 16, 32, 65, 177, 857, 8569, 678570;
...
Example: n-th row corresponds to P(n) = Pascal's triangle with 1's column shifted up 1 row, so that P(1) =
1;
1;
1, 1;
1, 2, 1;
1, 3, 3, 1;
...
then take lim_{k->infinity} (P(1))^k, getting A000110: (1, 1, 2, 5, 15, 52, ...), then delete the first 1.
Programs
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Sage
# generates the diagonals of the triangle, starting with diag = 1 the Bell numbers. def A171840_generator(len, diag) : A = [1]*diag for n in (0..len) : for k in range(n, 0, -1) : A[k - 1] += A[k] A.append(A[0]) yield A[0] for diag in (1..5) : print(list(A171840_generator(10, diag))) # Peter Luschny, Feb 27 2012
Formula
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.
Comments