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A158944 Triangle by columns: the natural numbers interleaved with zeros in every column: (1, 0, 2, 0, 3, 0, 4, ...) 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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

LINKS

Table of n, a(n) for n=1..78.

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

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)

EXAMPLE

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

MAPLE

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

CROSSREFS

Cf. A158943, A158945, A156663.

Sequence in context: A086713 A275730 A049771 * A156663 A139366 A049767

Adjacent sequences:  A158941 A158942 A158943 * A158945 A158946 A158947

KEYWORD

nonn,tabl,easy

AUTHOR

Gary W. Adamson, Mar 31 2009

STATUS

approved

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Last modified October 16 20:36 EDT 2021. Contains 348047 sequences. (Running on oeis4.)