A158944 Triangle by columns: the natural numbers interleaved with zeros in every column: (1, 0, 2, 0, 3, 0, 4, ...)

1, 0, 1, 2, 0, 1, 0, 2, 0, 1, 3, 0, 2, 0, 1, 0, 3, 0, 2, 0, 1, 4, 0, 3, 0, 2, 0, 1, 0, 4, 0, 3, 0, 2, 0, 1, 5, 0, 4, 0, 3, 0, 2, 0, 1, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1, 6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1, 0, 6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1

Offset: 1

Author

Gary W. Adamson, Mar 31 2009

Keywords

• nonn
• tabl
• easy

Comments

Eigensequence of the triangle = A158943: (1, 1, 3, 5, 10, 19, 36, 69, 131, ...)

Examples

			First few rows of the triangle =
  1;
  0, 1;
  2, 0, 1;
  0, 2, 0, 1;
  3, 0, 2, 0, 1;
  0, 3, 0, 2, 0, 1;
  4, 0, 3, 0, 2, 0, 1;
  0, 4, 0, 3, 0, 2, 0, 1;
  5, 0, 4, 0, 3, 0, 2, 0, 1;
  0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
  6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
  0, 6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
  7, 0, 6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
  ...
The inverse array begins
   1;
   0,  1;
  -2,  0,  1;
   0, -2,  0,  1;
   1,  0, -2,  0,  1;
   0,  1,  0, -2,  0,  1;
   0,  0,  1,  0, -2,  0,  1;
   0,  0,  0,  1,  0, -2,  0,  1;
   0,  0,  0,  0,  1,  0, -2,  0,  1;
   ... - _Peter Bala_, Aug 15 2021
		

Links

• D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012).

Crossrefs

Cf. A158943, A158945, A156663.

Programs

Maple

seq(seq((1/2)*(n - k + 2) * (1 + (-1)^(n-k))/2, k = 0..n), n = 0..10) # Peter Bala, Aug 15 2021

Formula

Triangle by columns: A027656: (1, 0, 2, 0, 3, 0, 4, 0, 5, ...) in every column.

From Peter Bala, Aug 15 2021: (Start)

T(n,k) = (1/2)*(n - k + 2) * (1 + (-1)^(n-k))/2 for 0 <= k <= n.

Double Riordan array (1/(1-x)^2; x, x) as defined in Davenport et al.

The m-th power of the array is the double Riordan array (1/(1 - x)^(2*m); x, x). Cf. A156663. (End)