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%I #21 Jul 26 2015 06:59:41
%S 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,
%T 1,7,52,266,424,392,224,64,1,8,79,526,1208,1368,1040,512,128,1,9,114,
%U 1079,3033,4432,4064,2656,1152,256
%N A Catalan-type triangle read by rows, generated by iteration of convolution squares.
%C 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.
%C 1, 1, 1, 1, 1, 1, 1, ...(given)
%C 1, 2, 3, 4, 5, 6, 7, ...
%C 1, 2, 5, 10, 18, 30, 47, ...
%C 1, 2, 5, 14, 34, 76, 161, ...
%C ...
%C where the rows converge to the Catalan numbers (A000108).
%C 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, ...).
%C 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:
%C 1;
%C 1, 1;
%C 1, 2, 2;
%C 1, 3, 6, 4;
%C 1, 4, 13, 16, 8;
%C 1, 5, 24, 46, 40, 16;
%C 1, 6, 40, 114, 140, 96, 32;
%C 1, 7, 52, 266, 424, 392, 224, 64;
%C ...
%e 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, ...).
%Y Cf. A000108 (row sums).
%K nonn,tabl
%O 1,5
%A _Gary W. Adamson_, Jul 05 2015
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