A259824 A Catalan-type triangle read by rows, generated by iteration of convolution squares.

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

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

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: A091187 A318607 A340106 * A065173 A330965 A098474

Adjacent sequences: A259821 A259822 A259823 * A259825 A259826 A259827

Keyword

nonn,tabl

Author

Gary W. Adamson, Jul 05 2015

Status

approved