A116855 Triangle read by rows, constructed from binomial transforms of prefixes of A000255 (see Comments for precise definition).

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

Offset

1,5

Comments

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.

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

The first few rows of S are

1, 1, 1, 1, ...

1, 2, 3, 4, ...

1, 2, 6, 13, ...

1, 2, 6, 24, ...

...

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

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, 67, 36, 7, 1;

...

Rows converge to sequence A000142 of factorial numbers.

Links

Table of n, a(n) for n=1..66.

Crossrefs

Cf. A000255, A010027.

Sequence in context: A179749 A103462 A225641 * A173265 A157744 A335545

Adjacent sequences: A116852 A116853 A116854 * A116856 A116857 A116858

Keyword

nonn,tabl

Author

Gary W. Adamson, Feb 24 2006

Extensions

Edited by N. J. A. Sloane, Aug 11 2019

Status

approved