A116855 - OEIS

%I #18 May 04 2024 09:24:15

%S 1,1,1,1,2,1,1,2,3,1,1,2,6,4,1,1,2,6,13,5,1,1,2,6,24,23,6,1,1,2,6,24,

%T 67,36,7,1,1,2,6,24,120,146,52,8,1,1,2,6,24,120,411,272,71,9,1,1,2,6,

%U 24,120,720,1067,456,93,10,1

%N Triangle read by rows, constructed from binomial transforms of prefixes of A000255 (see Comments for precise definition).

%C Form a matrix R whose columns are [1,0,0,0,...], [1,1,0,0,0,...], [1,1,3,0,0,0,...], [1,1,3,11,0,0,0,...], [1,1,3,11,53,0,0,0,...], ..., formed from the prefixes of A000255 each padded with infinitely many zeros.

%C Replace each column of R by its binomial transform to get the matrix S.

%C Rows converge to sequence A000142 of factorial numbers.

%e The first few rows of S are

%e   1, 1, 1,  1, ...

%e   1, 2, 3,  4, ...

%e   1, 2, 6, 13, ...

%e   1, 2, 6, 24, ...

%e   ...

%e Read S by antidiagonals to get the desired triangle, which begins

%e   1;

%e   1, 1;

%e   1, 2, 1;

%e   1, 2, 3,  1;

%e   1, 2, 6,  4,  1;

%e   1, 2, 6, 13,  5,  1;

%e   1, 2, 6, 24, 23,  6, 1;

%e   1, 2, 6, 24, 67, 36, 7, 1;

%e   ...

%Y Cf. A000255, A010027.

%K nonn,tabl

%O 1,5

%A _Gary W. Adamson_, Feb 24 2006

%E Edited by _N. J. A. Sloane_, Aug 11 2019