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

0,2

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.

Links

Table of n, a(n) for n=0..58.

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;

Example

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;

Crossrefs

Sequence in context: A124730 A114283 A106187 * A114423 A335109 A179863

Adjacent sequences: A110132 A110133 A110134 * A110136 A110137 A110138

Keyword

easy,nonn,tabl

Author

Paul Barry, Jul 13 2005

Status

approved