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

0,5

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

Links

Table of n, a(n) for n=0..90.

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.

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

Example

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;

Crossrefs

A signed version is A110509.

Sequence in context: A336087 A204387 A110509 * A319574 A204040 A325773

Adjacent sequences: A113950 A113951 A113952 * A113954 A113955 A113956

Keyword

easy,nonn,tabl

Author

Paul Barry, Nov 09 2005

Status

approved