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

1,3

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, ...).

Links

Table of n, a(n) for n=1..66.

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.

Example

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.

Prog

(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

Crossrefs

Cf. A007318, A007476, A171841, A000110.

Sequence in context: A308947 A175011 A211700 * A132309 A144224 A122881

Adjacent sequences: A171837 A171838 A171839 * A171841 A171842 A171843

Keyword

nonn,tabl

Author

Gary W. Adamson, Dec 19 2009

Status

approved