A113953 A Jacobsthal triangle.

1, 0, 1, 0, 2, 1, 0, 0, 4, 1, 0, 0, 4, 6, 1, 0, 0, 0, 12, 8, 1, 0, 0, 0, 8, 24, 10, 1, 0, 0, 0, 0, 32, 40, 12, 1, 0, 0, 0, 0, 16, 80, 60, 14, 1, 0, 0, 0, 0, 0, 80, 160, 84, 16, 1, 0, 0, 0, 0, 0, 32, 240, 280, 112, 18, 1, 0, 0, 0, 0, 0, 0, 192, 560, 448, 144, 20, 1, 0, 0, 0, 0, 0, 0, 64, 672, 1120, 672, 180, 22, 1

Offset: 0

Author

Paul Barry, Nov 09 2005

Keywords

• easy
• nonn
• tabl

Comments

Rows sums are the Jacobsthal numbers A001045(n+1).

Antidiagonal sums are the Padovan-Jacobsthal numbers A052947.

Inverse is (1,xc(-2x)), c(x) the g.f. of A000108, with general term k*C(2n-k-1,n-k)(-2)^(n - k)/n.

Triangle read by rows given by (0, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 01 2013

Examples

			Rows begin
  1;
  0,  1;
  0,  2,  1;
  0,  0,  4,  1;
  0,  0,  4,  6,  1;
  0,  0,  0, 12,  8,  1;
  0,  0,  0,  8, 24, 10,  1;
		

Links

• D. Merlini, R. Sprugnoli, M. C. Verri, Strip tiling and regular grammars, Theor. Comp. Sci 242 (1-2) (2000) 109-124, Table 1, p=2.

Crossrefs

A signed version is A110509.

Formula

G.f.: 1/(1-xy(1+2x)).

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

T(n,k) = 2^(n-k)*binomial(k, n-k).

T(n,k) = A026729(n,k)*2^(n-k). - Philippe Deléham, Nov 22 2006

T(n,k) = T(n-1,k-1) + 2*T(n-2,k-1), T(0,0) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Nov 01 2013