A093658 - OEIS

%I #6 Oct 24 2017 12:56:33

%S 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,

%T 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,

%U 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

%N 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].

%C Related to factorials, the incomplete gamma function (A010842) and the total number of arrangements of sets (A000522).

%C First column forms A093659, where A093659(2^n) = n! for n>=0.

%C Row sums form A093660, where A093660(2^n) = A000522(n) for n>=0.

%C Partial sums of the row sums form A093661, where A093661(2^n) = A010842(n) for n>=0.

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

%e Let M(n) be the lower triangular matrix formed from the first 2^n rows.

%e To generate M(3) from M(2), take the matrix square of M(2):

%e [1,0,0,0]^2=[1,0,0,0]

%e [1,1,0,0]...[2,1,0,0]

%e [1,0,1,0]...[2,0,1,0]

%e [2,1,1,1]...[6,2,2,1]

%e and append M(2)^2 to the bottom left corner and M(2) to the bottom right:

%e [1],

%e [1,1],

%e [1,0,1],

%e [2,1,1,1],

%e .........

%e [1,0,0,0],[1],

%e [2,1,0,0],[1,1],

%e [2,0,1,0],[1,0,1],

%e [6,2,2,1],[2,1,1,1].

%e Repeating this process converges to triangle A093658.

%Y Cf. A000522, A010842, A093655, A093662.

%K nonn,tabl

%O 1,7

%A _Paul D. Hanna_, Apr 08 2004