A124725 Triangle read by rows: T(n,k) = binomial(n,k) + binomial(n,k+2) (0 <= k <= n).

1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 7, 8, 7, 4, 1, 11, 15, 15, 11, 5, 1, 16, 26, 30, 26, 16, 6, 1, 22, 42, 56, 56, 42, 22, 7, 1, 29, 64, 98, 112, 98, 64, 29, 8, 1, 37, 93, 162, 210, 210, 162, 93, 37, 9, 1, 46, 130, 255, 372, 420, 372, 255, 130, 46, 10, 1, 56, 176, 385, 627, 792, 792, 627

Offset: 0

Author

Gary W. Adamson and Roger L. Bagula, Nov 05 2006

Keywords

• nonn
• tabl

Comments

Binomial transform of the infinite tridiagonal matrix with main diagonal, (1,1,1,...), subdiagonal, (0,0,0,...) and subsubdiagonal, (1,1,1,...). Sum of entries in row n = 2^(n+1) - n - 1 = A000325(n+1).

Riordan array ((1-2x+2x^2)/(1-x)^3, x/(1-x)). - Paul Barry, Apr 08 2011

Examples

			Row 3 = (4, 4, 3, 1), then row 4 = (7, 8, 7, 4, 1).
First few rows of the triangle are
   1;
   1,  1;
   2,  2,  1;
   4,  4,  3,  1;
   7,  8,  7,  4,  1;
  11, 15, 15, 11,  5,  1;
  16, 26, 30, 26, 16,  6,  1;
  ...
From _Paul Barry_, Apr 08 2011: (Start)
Production matrix begins
   1, 1;
   1, 1, 1;
   0, 0, 1, 1;
  -1, 0, 0, 1, 1;
   0, 0, 0, 0, 1, 1;
   1, 0, 0, 0, 0, 1, 1;
   0, 0, 0, 0, 0, 0, 1, 1;
  -1, 0, 0, 0, 0, 0, 0, 1, 1;
   0, 0, 0, 0, 0, 0, 0, 0, 1, 1;
   1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1;
(End)
		

Crossrefs

Cf. A098574, A000125, A055795, A027660, A055796, A002663.

Cf. A000325.

Programs

Maple

T:=(n,k)->binomial(n,k)+binomial(n,k+2): for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Mathematica

Flatten[Table[Binomial[n,k]+Binomial[n,k+2],{n,0,20},{k,0,n}]] (* Harvey P. Dale, Jun 12 2015 *)

Formula

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 3*T(n-2,k) - 2*T(n-2,k-1) + T(n-3,k) + T(n-3,k-1), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0) = T(2,1) = 2, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Feb 12 2014

Extensions

Edited by N. J. A. Sloane, Nov 29 2006