A293171 Triangle read by rows: T(n,k) = number of colored weighted Motzkin paths ending at (n,k).

1, 1, 1, 9, 2, 1, 25, 15, 3, 1, 145, 52, 22, 4, 1, 561, 285, 90, 30, 5, 1, 2841, 1206, 495, 140, 39, 6, 1, 12489, 6027, 2261, 791, 203, 49, 7, 1, 60705, 27560, 11452, 3864, 1190, 280, 60, 8, 1, 281185, 134073, 54468, 20076, 6174, 1710, 372, 72, 9, 1, 1353769, 633130, 268845, 99240, 33090, 9372, 2370, 480, 85, 10, 1

Offset: 0

Author

N. J. A. Sloane, Oct 19 2017

Keywords

• nonn
• tabl

Examples

			Triangle begins:
1,
1,1,
9,2,1,
25,15,3,1,
145,52,22,4,1,
561,285,90,30,5,1,
...
		

Links

• 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. 3087.

Crossrefs

First column is A084605, 2nd A098520.

Programs

Maple

A293171 := proc(n,k)
    option remember;
    local b,e,c;
    b := 1; e:= 2; c := e^2 ;
    if k < 0 or k > n then
        0;
    elif k = n then
        1;
    elif k = 0 then
        b*procname(n-1,0)+2*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( A293171(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Oct 27 2017

Mathematica

T[n_, k_] := T[n, k] = Module[{b=1, e=2, c=4}, Which[k<0 || k>n, 0, k==n, 1, k == 0, b*T[n-1, 0] + 2*c*T[n-1, 1], True, T[n-1, k-1] + b*T[n-1, k] + c*T[n-1, k+1]]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 19 2019, after R. J. Mathar *)