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

0,4

Links

Table of n, a(n) for n=0..65.

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.

Example

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

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 *)

Crossrefs

First column is A084605, 2nd A098520.

Sequence in context: A293258 A010536 A239908 * A334689 A335086 A151898

Adjacent sequences: A293168 A293169 A293170 * A293172 A293173 A293174

Keyword

nonn,tabl

Author

N. J. A. Sloane, Oct 19 2017

Status

approved