A161869 Convergent of an infinite product of Pascal's triangles aerated by rows.

1, 1, 2, 4, 8, 16, 33, 71, 160, 376, 912, 2256, 5633, 14093, 35170, 87344, 215680, 529568, 1293633, 3146515, 7627208, 18441476, 44510160, 107310480, 258566402, 622900466, 1500717220, 3616471960, 8717948688, 21023129472, 50713990918, 122374025914, 295366777856

Offset: 0

Author

Gary W. Adamson, Jun 20 2009

Keywords

• nonn
• easy

Comments

The sequence may be the binomial transform of A024493 interleaved with zeros. A024493 = (1, 1, 1, 2, 5, 11, 22,...); so the conjecture succeeds through a(12) = A007318 * [1, 0, 1, 0, 1, 0, 2, 0, 5, 0, 11,...].

Calculating more terms of the sequence shows the above conjecture is incorrect. - Peter Bala, Jul 07 2015

Examples

			Pascal(1)     Pascal(2)    Pascal(3)
1              1            1
1 1            0 0          0 0
1 2 1          1 1 0        0 0 0
1 3 3 1        0 0 0 0      1 1 0 0
1 4 6 4 1      1 2 1 0 0    0 0 0 0 0
...
First columns of
Pascal(1):
1, 1, 1, 1, 1, 1, 1, 1,...
Pascal(1)*Pascal(2):
1, 1, 2, 4, 8, 16, 32, 64,...
Pascal(1)*Pascal(2)*Pascal(3):
1, 1, 2, 4, 8, 16, 33, 71,...
Pascal(1)*Pascal(2)*Pascal(3)*Pascal(4):
1, 1, 2, 4, 8, 16, 33, 71,...
...
converging to A161869.
		

Crossrefs

Cf. A024493, A027826 (from first column of Pascal(2)^n as n -> inf).

Programs

Maple

#A161869
#define aerated Pascal matrices (note indexing starts at 1)
Pascal := proc (n) local i, j, r;
Matrix(33, 33, (i, j) -> (product(r-(mod(i-1, n)), r = 1 .. n-1))*binomial(floor((i-1)/n), j-1) )/factorial(n-1) end proc:
#it suffices to take the product of the first four aerated
#Pascal arrays to get 33 correct terms of the sequence
seq((Pascal(1).Pascal(2).Pascal(3).Pascal(4))(n, 1), n = 1 .. 33);
# Peter Bala, Jul 07 2015

Formula

From Peter Bala, Jul 07 2015: (Start)

Construct an infinite set of Pascal's triangles aerated by rows, denoted Pascal(1), Pascal(2), ..., where Pascal(1) = A007318, Pascal(2) is an aerated version of Pascal(1) with alternate rows 1, 3, 5, ... set equal to (0, 0, 0, ...), Pascal(3) is a further aeration of Pascal(1) with now two adjacent rows set equal to (0, 0, 0, ...), and so on.

The infinite product Pascal(1)*Pascal(2)*Pascal(3)*... is well-defined. This sequence is the first column of the infinite product - all the other entries in the product are zero. (End)

Extensions

Terms a(12) through a(32) added by Peter Bala, Jul 07 2015