A130128 Triangle read by rows: T(n,k) = (n - k + 1)*2^(k-1).

1, 2, 2, 3, 4, 4, 4, 6, 8, 8, 5, 8, 12, 16, 16, 6, 10, 16, 24, 32, 32, 7, 12, 20, 32, 48, 64, 64, 8, 14, 24, 40, 64, 96, 128, 128, 9, 16, 28, 48, 80, 128, 192, 256, 256, 10, 18, 32, 56, 96, 160, 256, 384, 512, 512, 11, 20, 36, 64, 112, 192, 320, 512, 768, 1024, 1024

Offset

1,2

Comments

T(n,k) is the number of paths from node 0 to odd k in a directed graph with 2n+1 vertices labeled 0, 1, ..., 2n+1 and edges leading from i to i+1 for all i, from i to i+2 for even i, and from i to i-2 for odd i. - Grace Work, Mar 01 2020

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275

E. Krom and M. M. Roughan, Path Counting and Eulerian Numbers, Girls' Angle Bulletin, Vol. 13, No. 3 (2020), 8-10.

Formula

Equals A004736 * A130123 as infinite lower triangular matrices.

As a square array, n >= 0, k >= 1, read by descending antidiagonals, A(n,k) = k * 2^n. - Peter Munn, Sep 22 2022

G.f.: x*y/( (1-x)^2 * (1-2*x*y) ). - Kevin Ryde, Sep 24 2022

Example

First few rows of the triangle are:
  1;
  2,  2;
  3,  4,  4;
  4,  6,  8,  8;
  5,  8, 12, 16, 16;
  6, 10, 16, 24, 32, 32;
  7, 12, 20, 32, 48, 64, 64;
  ...
From Peter Munn, Sep 22 2022: (Start)
As a square array, showing top left:
    1,   2,   3,    4,    5,    6,    7, ...
    2,   4,   6,    8,   10,   12,   14, ...
    4,   8,  12,   16,   20,   24,   28, ...
    8,  16,  24,   32,   40,   48,   56, ...
   16,  32,  48,   64,   80,   96,  112, ...
   32,  64,  96,  128,  160,  192,  224, ...
  ...
(End)

Mathematica

Table[(n - k + 1)*2^(k - 1), {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, Mar 23 2020 *)

Prog

(PARI) T(n, k)={(n - k + 1)*2^(k-1)} \\ Andrew Howroyd, Mar 01 2020

Crossrefs

Row sums are A000295.

Cf. A004736, A054582 (subtable of square array), A130123.

Sequence in context: A367410 A259197 A309559 * A210556 A208914 A049980

Adjacent sequences: A130125 A130126 A130127 * A130129 A130130 A130131

Keyword

nonn,easy,tabl,walk

Author

Gary W. Adamson, May 11 2007

Extensions

Name clarified by Grace Work, Mar 01 2020

Terms a(56) and beyond from Andrew Howroyd, Mar 01 2020

Status

approved