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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->inf.}, k=1,2,3,...; (P(n))^k, deleting the first 1. 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 (list; table; graph; refs; listen; history; text; internal format)
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->inf.} (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=1..inf.} (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. A007476, A171841, A000110

Sequence in context: A002211 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

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Last modified March 20 15:43 EDT 2019. Contains 321345 sequences. (Running on oeis4.)