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

Author

Gary W. Adamson, Dec 19 2009

Keywords

• nonn
• tabl

Comments

Row sums = A171841: (1, 3, 8, 22, 68, 241, 974, ...).

Right border = the Bell sequence A000110 starting (1, 2, 5, 15, 52, ...).

Row 2 of the array = A007476 starting (1, 1, 2, 4, 9, 23, 65, 199, ...).

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.
		

Crossrefs

Cf. A007318, A007476, A171841, A000110.

Programs

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.