A098520 -id:A098520 - cp's OEIS frontend

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

N. J. A. Sloane, Oct 19 2017

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

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.

First column is A084605, 2nd A098520.

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

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 *)

A272868 Triangle read by rows, T(n,k) = 2^k*GegenbauerC(k,-n,-1/4), for n>=0 and 0<=k<=n.

Original entry on oeis.org

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

Offset: 0

Peter Luschny, May 08 2016

			Triangle starts:
                                1;
                              1, 1;
                            1, 2, 9;
                          1, 3, 15, 25;
                       1, 4, 22, 52, 145;
                     1, 5, 30, 90, 285, 561;
                 1, 6, 39, 140, 495, 1206, 2841;
             1, 7, 49, 203, 791, 2261, 6027, 12489;

Cf. A084605, A091147, A098520.

T := (n,k) -> simplify(2^k*GegenbauerC(k, -n, -1/4)):
for n from 0 to 9 do seq(T(n,k), k=0..n) od;

Table[If[n == 0, 1, 2^k GegenbauerC[k, -n, -1/4]], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, May 08 2016 *)

T(n,n) = A084605(n).
T(n,n-1) = A098520(n).
T(n+1,n)/(n+1) = A091147(n).

cp's OEIS Frontend

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

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