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A161869
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Convergent of an infinite product of Pascal's triangles aerated by rows.
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2
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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
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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FORMULA
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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)
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EXAMPLE
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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,...
...
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MAPLE
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#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);
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Terms a(12) through a(32) added by Peter Bala, Jul 07 2015
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STATUS
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approved
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