A380128 Triangle read by rows: Riordan array (1/(C(x)*sqrt(1-4*x)), x/C(x)) where C(x) is g.f. of A000108.

1, 1, 1, 3, 0, 1, 10, 1, -1, 1, 35, 4, 0, -2, 1, 126, 15, 1, 0, -3, 1, 462, 56, 5, 0, 1, -4, 1, 1716, 210, 21, 1, 0, 3, -5, 1, 6435, 792, 84, 6, 0, 0, 6, -6, 1, 24310, 3003, 330, 28, 1, 0, -1, 10, -7, 1, 92378, 11440, 1287, 120, 7, 0, 0, -4, 15, -8, 1, 352716, 43758, 5005, 495, 36, 1, 0, 0, -10, 21, -9, 1

Offset: 0

Author

Werner Schulte, Jan 12 2025

Keywords

• sign
• easy
• tabl

Examples

			Triangle T(n, k) for 0 <= k <= n starts:
n \k :       0      1     2    3   4   5   6   7    8   9  10  11
=================================================================
   0 :       1
   1 :       1      1
   2 :       3      0     1
   3 :      10      1    -1    1
   4 :      35      4     0   -2   1
   5 :     126     15     1    0  -3   1
   6 :     462     56     5    0   1  -4   1
   7 :    1716    210    21    1   0   3  -5   1
   8 :    6435    792    84    6   0   0   6  -6    1
   9 :   24310   3003   330   28   1   0  -1  10   -7   1
  10 :   92378  11440  1287  120   7   0   0  -4   15  -8   1
  11 :  352716  43758  5005  495  36   1   0   0  -10  21  -9   1
  etc.
		

Crossrefs

Cf. A088218 (column 0), A001791 (column 1), A000007 (central terms).

Cf. A000108, A001764.

Programs

PARI

T(n, k) = polcoef(polcoef(1/((C(t)-x*t)*sqrt(1-4*t))+x*O(x^k), k, x)+t*O(t^n), n, t);
       m = matrix(15, 15, n, k, if(k>n, 0, T(n-1, k-1)))

Formula

G.f.: 1/((C(t) - x*t) * sqrt(1 - 4*t)) where C(t) is g.f. of A000108.

Inverse Riordan array is (2 - D(x), x * D(x)) where D(x) is g.f. of A001764.

Conjecture: T(n, k) = Sum_{i=0..n-k} (-1)^i * binomial(2*n - 2*k, n - k - i) * binomial(k + i, i).