A113214 Riordan array (1+2x,x(1+x)).

1, 2, 1, 0, 3, 1, 0, 2, 4, 1, 0, 0, 5, 5, 1, 0, 0, 2, 9, 6, 1, 0, 0, 0, 7, 14, 7, 1, 0, 0, 0, 2, 16, 20, 8, 1, 0, 0, 0, 0, 9, 30, 27, 9, 1, 0, 0, 0, 0, 2, 25, 50, 35, 10, 1, 0, 0, 0, 0, 0, 11, 55, 77, 44, 11, 1, 0, 0, 0, 0, 0, 2, 36, 105, 112, 54, 12, 1, 0, 0, 0, 0, 0, 0, 13, 91, 182, 156, 65, 13, 1

Offset: 0

Author

Paul Barry, Oct 18 2005

Keywords

• easy
• nonn
• tabl

Comments

Row sums are Lucas numbers A000204. Diagonal sums are A007307(n+1). Inverse is (-1)^(n-k)A092392(n,k). Product with Pascal triangle (1/(1-x),x/(1-x)) is A111125.

Examples

			Triangle begins
  1;
  2,  1;
  0,  3,  1;
  0,  2,  4,  1;
  0,  0,  5,  5,  1;
  0,  0,  2,  9,  6,  1;
  0,  0,  0,  7, 14,  7,  1;
  0,  0,  0,  2, 16, 20,  8,  1;
Row 4: (1 + x*c(-x))^5 = 1 + 5*x + 5*x^2 + O(x^5). - _Peter Bala_, Sep 10 2021
		

Links

• R. Tauraso, Edge cover time for regular graphs, JIS 11 (2008) 08.4.4.

Crossrefs

Cf. A000108, A000204, A007307, A092392, A111125.

Formula

T(n, k) = C(k, n-k) + 2*C(k, n-k-1).

T(n, k) = Sum_{j = 0..n} (-1)^(n-j)*C(n, j)*C(j+k, 2*k)*(2*j+1)/(2*k+1).

From Peter Bala, Sep 10 2021: (Start)

T(n,k) = T(n-1,k-1) + T(n-2,k-1) with boundary conditions T(n,n) = 1, T(1,0) = 2 and T(n,k) = 0 for k < 0 or k > n.

The entries in row n, read in reverse order, are the coefficients in the n-th degree Taylor polynomial of (1 + x*c(-x))^(n+1) at x = 0, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)