A093658 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].

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

Offset: 1

Author

Paul D. Hanna, Apr 08 2004

Keywords

• nonn
• tabl

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.

Examples

			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.
		

Crossrefs

Cf. A000522, A010842, A093655, A093662.

Formula

T(2^n, 1) = n! for n>=0.