A110135 Square array of expansions of 1/sqrt(1-4x-4*k*x^2), read by antidiagonals.

1, 2, 1, 6, 2, 1, 20, 8, 2, 1, 70, 32, 10, 2, 1, 252, 136, 44, 12, 2, 1, 924, 592, 214, 56, 14, 2, 1, 3432, 2624, 1052, 304, 68, 16, 2, 1, 12870, 11776, 5284, 1632, 406, 80, 18, 2, 1, 48620, 53344, 26840, 9024, 2332, 520, 92, 20, 2, 1, 184756, 243392, 137638, 50304

Offset: 0

Author

Paul Barry, Jul 13 2005

Keywords

• easy
• nonn
• tabl

Comments

Column k has g.f. 1/sqrt(1-4x-4*k*x^2) and e.g.f. exp(2x)BesselI(0,2*sqrt(k)x). Columns include A000984, A006139, A084609, A098453. Row sums of triangle are A110136. Diagonal sums of triangle are A110137.

Examples

			As a square array, rows start
    1,   1,    1,    1,    1, ...
    2,   2,    2,    2,    2, ...
    6,   8,   10,   12,   14,   16, ...
   20,  32,   44,   56,   68,   80, ...
   70, 136,  214,  304,  406,  520, ...
  252, 592, 1052, 1632, 2332, 3152, ...
As a number triangle, rows start
    1;
    2,   1;
    6,   2,   1;
   20,   8,   2,   1;
   70,  30,  10,   2,   1;
  252, 136,  44,  12,   2,   1;
		

Formula

Square array T(n, k) = Sum_{j=0..floor(n/2)} C(n, j)*C(2(n-j), n)*k^j.

Number triangle T1(n, k) = Sum_{j=0..floor((n-k)/2)} C(n-k, j)*C(2(n-k-j), n-k)*k^j;