A156663 Triangle by columns, powers of 2 interleaved with zeros.

1, 0, 1, 2, 0, 1, 0, 2, 0, 1, 4, 0, 2, 0, 1, 0, 4, 0, 2, 0, 1, 8, 0, 4, 0, 2, 0, 1, 0, 8, 0, 4, 0, 2, 0, 1, 16, 0, 8, 0, 4, 0, 2, 0, 1, 0, 16, 0, 8, 0, 4, 0, 2, 0, 1, 32, 0, 16, 0, 8, 0, 4, 0, 2, 0, 1, 0, 32, 0, 16, 0, 8, 0, 4, 0, 2, 0, 1

Offset: 0

Author

Gary W. Adamson, Feb 12 2009

Keywords

• nonn
• tabl

Comments

Eigensequence of the triangle = A001045.

Examples

			First few rows of the triangle =
   1;
   0,  1;
   2,  0,  1;
   0,  2,  0,  1;
   4,  0,  2,  0,  1;
   0,  4,  0,  2,  0,  1;
   8,  0,  4,  0,  2,  0,  1;
   0,  8,  0,  4,  0,  2,  0,  1;
  16,  0,  8,  0,  4,  0,  2,  0,  1;
   0, 16,  0,  8,  0,  4,  0,  2,  0,  1;
  32,  0, 16,  0,  8,  0,  4,  0,  2,  0,  1;
   0, 32,  0, 16,  0,  8,  0,  4,  0,  2,  0,  1;
  ...
The inverse array begins
   1;
   0,  1;
  -2,  0,  1;
   0, -2,  0,  1;
   0,  0, -2,  0,  1;
   0,  0,  0, -2,  0,  1;
   0,  0,  0,  0, -2,  0,  1;
   0,  0,  0,  0,  0, -2,  0,  1;
   0,  0,  0,  0,  0,  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. A001045, A158944.

Programs

Maple

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

Formula

Triangle by columns, (1, 0, 2, 0, 4, 0, 8, ...) in every column.

From Peter Bala, Aug 15 2021: (Start)

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

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

The m-th power of the array is the double Riordan array (1/(1 - 2*x^2)^(m); x, x). Cf. A158944. (End)

Extensions

Typo in Data corrected by Peter Bala, Aug 15 2021