

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 nth 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
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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 nth 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: nth 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



