|
|
A259824
|
|
A Catalan-type triangle read by rows, generated by iteration of convolution squares.
|
|
0
|
|
|
1, 1, 1, 1, 2, 2, 1, 3, 6, 4, 1, 4, 13, 16, 8, 1, 5, 24, 46, 40, 16, 1, 6, 40, 114, 140, 96, 32, 1, 7, 52, 266, 424, 392, 224, 64, 1, 8, 79, 526, 1208, 1368, 1040, 512, 128, 1, 9, 114, 1079, 3033, 4432, 4064, 2656, 1152, 256
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,5
|
|
COMMENTS
|
The triangle is generated by an iterative procedure in which the (n+1)-st row of the following array is the convolution square of the n-th row prepended with a 1.
1, 1, 1, 1, 1, 1, 1, ...(given)
1, 2, 3, 4, 5, 6, 7, ...
1, 2, 5, 10, 18, 30, 47, ...
1, 2, 5, 14, 34, 76, 161, ...
...
where the rows converge to the Catalan numbers (A000108).
Example: second row of (1, 2, 3, ...) is prepended with a 1: (1, 1, 2, 3, ...), and the convolution square of that sequence is row 3 (1, 2, 5, 10, 18, ...).
Next, take finite differences of terms by columns; such that the finite difference row of the n-th column becomes the n-th row of the triangle. First few rows of the triangle (as an infinite lower triangular matrix with the rest zeros) are:
1;
1, 1;
1, 2, 2;
1, 3, 6, 4;
1, 4, 13, 16, 8;
1, 5, 24, 46, 40, 16;
1, 6, 40, 114, 140, 96, 32;
1, 7, 52, 266, 424, 392, 224, 64;
...
|
|
LINKS
|
|
|
EXAMPLE
|
Row 4 of the triangle is (1, 3, 6, 4, 0, 0, 0, ...) since the finite differences of row 4 of the array (1, 4, 10, 14, 14, ...) are (1, 3, 6, 4, 0, 0, 0, ...).
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|