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

Author

Gary W. Adamson, May 11 2007

Keywords

• nonn
• easy
• tabl
• walk

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

Examples

			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)
		

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.

Crossrefs

Row sums are A000295.

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

Programs

Mathematica

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

PARI

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

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

Extensions

Name clarified by Grace Work, Mar 01 2020

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