|
%I #25 Sep 24 2022 19:49:22
%S 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,
%T 64,8,14,24,40,64,96,128,128,9,16,28,48,80,128,192,256,256,10,18,32,
%U 56,96,160,256,384,512,512,11,20,36,64,112,192,320,512,768,1024,1024
%N Triangle read by rows: T(n,k) = (n - k + 1)*2^(k-1).
%C 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
%H Andrew Howroyd, <a href="/A130128/b130128.txt">Table of n, a(n) for n = 1..1275</a>
%H E. Krom and M. M. Roughan, <a href="http://girlsangle.org/page/bulletin-archive/GABv13n03E.pdf">Path Counting and Eulerian Numbers</a>, Girls' Angle Bulletin, Vol. 13, No. 3 (2020), 8-10.
%F Equals A004736 * A130123 as infinite lower triangular matrices.
%F As a square array, n >= 0, k >= 1, read by descending antidiagonals, A(n,k) = k * 2^n. - _Peter Munn_, Sep 22 2022
%F G.f.: x*y/( (1-x)^2 * (1-2*x*y) ). - _Kevin Ryde_, Sep 24 2022
%e First few rows of the triangle are:
%e 1;
%e 2, 2;
%e 3, 4, 4;
%e 4, 6, 8, 8;
%e 5, 8, 12, 16, 16;
%e 6, 10, 16, 24, 32, 32;
%e 7, 12, 20, 32, 48, 64, 64;
%e ...
%e From _Peter Munn_, Sep 22 2022: (Start)
%e As a square array, showing top left:
%e 1, 2, 3, 4, 5, 6, 7, ...
%e 2, 4, 6, 8, 10, 12, 14, ...
%e 4, 8, 12, 16, 20, 24, 28, ...
%e 8, 16, 24, 32, 40, 48, 56, ...
%e 16, 32, 48, 64, 80, 96, 112, ...
%e 32, 64, 96, 128, 160, 192, 224, ...
%e ...
%e (End)
%t Table[(n - k + 1)*2^(k - 1), {n, 11}, {k, n}] // Flatten (* _Michael De Vlieger_, Mar 23 2020 *)
%o (PARI) T(n,k)={(n - k + 1)*2^(k-1)} \\ _Andrew Howroyd_, Mar 01 2020
%Y Row sums are A000295.
%Y Cf. A004736, A054582 (subtable of square array), A130123.
%K nonn,easy,tabl,walk
%O 1,2
%A _Gary W. Adamson_, May 11 2007
%E Name clarified by _Grace Work_, Mar 01 2020
%E Terms a(56) and beyond from _Andrew Howroyd_, Mar 01 2020
|
