

A259824


A Catalantype 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
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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 nth 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 nth column becomes the nth 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

Table of n, a(n) for n=1..55.


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

Cf. A000108 (row sums).
Sequence in context: A082137 A091187 A318607 * A065173 A098474 A153199
Adjacent sequences: A259821 A259822 A259823 * A259825 A259826 A259827


KEYWORD

nonn,tabl


AUTHOR

Gary W. Adamson, Jul 05 2015


STATUS

approved



