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A093658
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Lower triangular matrix, read by rows, defined as the convergent of the concatenation of matrices using the iteration: M(n+1) = [[M(n),0*M(n)],[M(n)^2,M(n)]], with M(0) = [1].
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7
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1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 1, 1, 2, 0, 1, 0, 1, 0, 1, 6, 2, 2, 1, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 1, 1, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 6, 2, 2, 1, 0, 0, 0, 0, 2, 1, 1, 1, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 6, 2, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 1, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,7
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COMMENTS
| Related to factorials, the incomplete gamma function (A010842) and the total number of arrangements of sets (A000522). First column forms A093659, where A093659(2^n) = n! for n>=0. Row sums form A093660, where A093660(2^n) = A000522(n) for n>=0. Partial sums of the row sums form A093661, where A093661(2^n) = A010842(n) for n>=0.
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FORMULA
| T(2^n, 1) = n! for n>=0.
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EXAMPLE
| Let M(n) be the lower triangular matrix formed from the first 2^n rows.
To generate M(3) from M(2), take the matrix square of M(2):
[1,0,0,0]^2=[1,0,0,0]
[1,1,0,0]...[2,1,0,0]
[1,0,1,0]...[2,0,1,0]
[2,1,1,1]...[6,2,2,1]
and append M(2)^2 to the bottom left corner and M(2) to the bottom right:
[1],
[1,1],
[1,0,1],
[2,1,1,1],
.........
[1,0,0,0],[1],
[2,1,0,0],[1,1],
[2,0,1,0],[1,0,1],
[6,2,2,1],[2,1,1,1].
Repeating this process converges to triangle A093658.
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CROSSREFS
| Cf. A000522, A010842, A093655, A093662.
Sequence in context: A056977 A085425 A167230 * A096493 A076882 A016397
Adjacent sequences: A093655 A093656 A093657 * A093659 A093660 A093661
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Apr 08 2004
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