A026714 Triangular array T read by rows: T(n,0)=T(n,n)=1 for n >= 0; for n >= 2 and 1<=k<=n-1, T(n,k)=T(n-1,k-1)+T(n-2,k-1)+T(n-1,k) if k=[ (n-1)/2 ] or k=[ n/2 ] or k=[ (n+2)/2 ], else T(n,k)=T(n-1,k-1)+T(n-1,k).

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 13, 7, 1, 1, 8, 25, 25, 8, 1, 1, 9, 40, 63, 40, 9, 1, 1, 10, 49, 128, 128, 49, 10, 1, 1, 11, 59, 217, 319, 217, 59, 11, 1, 1, 12, 70, 276, 664, 664, 276, 70, 12, 1, 1, 13, 82, 346, 1157, 1647, 1157, 346, 82, 13, 1

Offset

1,5

Links

Table of n, a(n) for n=1..66.

Formula

T(n, k) = number of paths from (0, 0) to (n-k, k) in the directed graph having vertices (i, j) and edges (i, j)-to-(i+1, j) and (i, j)-to-(i, j+1) for i, j >= 0 and edges (i, j)-to-(i+1, j+1) if |i-j|<=2.

Example

   1
   1    1
   1    3    1
   1    5    5    1
   1    7   13    7    1
   1    8   25   25    8    1
   1    9   40   63   40    9    1
   1   10   49  128  128   49   10    1
   1   11   59  217  319  217   59   11    1
   1   12   70  276  664  664  276   70   12    1
   1   13   82  346 1157 1647 1157  346   82   13    1
   1   14   95  428 1503 3468 3468 1503  428   95   14    1
   1   15  109  523 1931 6128 8583 6128 1931  523  109   15    1

Maple

A026714 := proc(n, k)
    option remember;
    if n < 0 or k < 0 then
        0;
    elif n =0 or n= k then
        1;
    elif k = floor((n-1)/2) or k = floor(n/2) or k = floor(n/2+1) then
        procname(n-1, k-1)+procname(n-2, k-1)+procname(n-1, k) ;
    else
        procname(n-1, k-1)+procname(n-1, k) ;
    end if;
end proc: # R. J. Mathar, Oct 21 2019

Mathematica

T[n_, k_] := T[n, k] = Which[n < 0 || k < 0, 0, n == 0 || n == k, 1, k == Floor[(n - 1)/2] || k == Floor[n/2] || k == Floor[n/2 + 1], T[n - 1, k - 1] + T[n - 2, k - 1] + T[n - 1, k], True, T[n - 1, k - 1] + T[n - 1, k]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 23 2020 *)

Crossrefs

Cf. A026721 (row sums).

Sequence in context: A103450 A128254 A277930 * A008288 A238339 A302997

Adjacent sequences: A026711 A026712 A026713 * A026715 A026716 A026717

Keyword

nonn,tabl

Author

Clark Kimberling

Status

approved