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A generalized Riordan array related to Hankel and Toeplitz+Hankel transforms.
1

%I #4 Apr 19 2015 16:01:32

%S 1,1,1,1,2,1,2,3,2,1,3,6,4,2,1,6,10,8,5,2,1,10,20,15,10,6,2,1,20,35,

%T 30,21,12,7,2,1,35,70,56,42,28,14,8,2,1,70,126,112,84,56,36,16,9,2,1,

%U 126,252,210,168,120,72,45,18,10,2,1

%N A generalized Riordan array related to Hankel and Toeplitz+Hankel transforms.

%C Corresponds to the Riordan array ((1+2x)/sqrt(1-4x^2),xc(x^2)), c(x) the g.f. of A000108, with the first column replaced by 1,1,1,2,3,6,... with g.f. (1+2x+sqrt(1-4x^2))/(2*sqrt(1-4x^2)). Alternatively it is the array with first column 1,1,1,2,3,6,... and then the Riordan array ((1+2x)c(x)/sqrt(1-4x^2),xc(x^2)) embedded from the (1,1) position (indexing starting at (0,0)). Row sums are 2^n. Images of common sequences under this array have interesting Hankel transforms. For instance, the image of r^n has Hankel transform with g.f. 1/(1+(r^2-1)x^2).

%H E.L. Basor, T. Erhardt, <a href="http://arxiv.org/abs/math/0008075">Some identities for determinants of structured matrices</a>, arXiv:math/0008075v1

%e Triangle begins

%e 1,

%e 1, 1,

%e 1, 2, 1,

%e 2, 3, 2, 1,

%e 3, 6, 4, 2, 1,

%e 6, 10, 8, 5, 2, 1,

%e 10, 20, 15, 10, 6, 2, 1,

%e 20, 35, 30, 21, 12, 7, 2, 1,

%e 35, 70, 56, 42, 28, 14, 8, 2, 1,

%e 70, 126, 112, 84, 56, 36, 16, 9, 2, 1,

%e 126, 252, 210, 168, 120, 72, 45, 18, 10, 2, 1

%K easy,nonn,tabl

%O 0,5

%A _Paul Barry_, Oct 23 2007