A293172 Triangle read by rows: T(n,k) = number of colored weighted Motzkin paths ending at (n,k).

1, 6, 1, 40, 10, 1, 280, 84, 14, 1, 2016, 672, 144, 18, 1, 14784, 5280, 1320, 220, 22, 1, 109824, 41184, 11440, 2288, 312, 26, 1, 823680, 320320, 96096, 21840, 3640, 420, 30, 1, 6223360, 2489344, 792064, 198016, 38080, 5440, 544, 34, 1, 47297536, 19348992, 6449664, 1736448, 372096, 62016, 7752

Offset

0,2

Links

Table of n, a(n) for n=0..51.

Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, Some matrix identities on colored Motzkin paths, Discrete Mathematics 340.12 (2017): 3081-3091. See p. 3088.

Example

Triangle begins:
1,
6,1,
40,10,1,
280,84,14,1,
2016,672,144,18,1,
14784,5280,1320,220,22,1,
...

Maple

A293172 := proc(n, k)
    option remember;
    local b, d, r, c, e;
    b := 4; d:= 2; r := 2 ; c := r^2 ; e := d ;
    if k < 0 or k > n then
        0;
    elif k = n then
        1;
    elif k = 0 then
        (b+e)*procname(n-1, 0)+c*procname(n-1, 1) ;
    else
        procname(n-1, k-1)+b*procname(n-1, k)+c*procname(n-1, k+1) ;
    end if;
end proc:
seq(seq( A293172(n, k), k=0..n), n=0..15) ; # R. J. Mathar, Oct 27 2017

Mathematica

T[n_, k_] := T[n, k] = Module[{b = 4, d = 2, r = 2, c, e}, c = r^2; e = d; If[k < 0 || k > n, 0, If[k == n, 1, If[k == 0, (b + e) T[n - 1, 0] + c T[n - 1, 1], T[n - 1, k - 1] + b T[n - 1, k] + c T[n - 1, k + 1]]]]];
Table[T[n, k], {n, 0, 15}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 07 2020, from Maple *)

Crossrefs

First column is A069720.

Sequence in context: A089504 A145927 A113365 * A145356 A145357 A035529

Adjacent sequences: A293169 A293170 A293171 * A293173 A293174 A293175

Keyword

nonn,tabl

Author

N. J. A. Sloane, Oct 19 2017

Status

approved