A026637 Triangular array T read by rows: T(n,0)=T(n,n)=1 for n >= 0; T(n,1)=T(n,n-1)=[ (3n-1)/2 ] for n >= 1; T(n,k)=T(n-1,k-1)+T(n-1,k) for 2<=k<=n-2, n >= 4.

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 5, 8, 5, 1, 1, 7, 13, 13, 7, 1, 1, 8, 20, 26, 20, 8, 1, 1, 10, 28, 46, 46, 28, 10, 1, 1, 11, 38, 74, 92, 74, 38, 11, 1, 1, 13, 49, 112, 166, 166, 112, 49, 13, 1, 1, 14, 62, 161, 278, 332, 278, 161, 62, 14, 1, 1, 16, 76

Offset

0,5

Comments

See A228053 for a sequence with many terms in common with this one. - T. D. Noe, Aug 07 2013

Links

Reinhard Zumkeller, Rows n = 0..100 of table, flattened

Formula

T(n, k) = number of paths from (0, 0) to (n-k, k) in 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) for i=0, j >= 1 and odd and for j=0, i >= 1 and odd.

Example

1;
1,1;
1,2,1;
1,4,4,1;
1,5,8,5,1;
1,7,13,13,7,1;
1,8,20,26,20,8,1;
1,10,28,46,46,28,10,1;
1,11,38,74,92,74,38,11,1;
1,13,49,112,166,166,112,49,13,1;
1,14,62,161,278,332,278,161,62,14,1;

Maple

A026637 := proc(n, k)
      option remember;
      if k=0 or k=n then
        1
    elif k=1 or k=n-1 then
        floor((3*n-1)/2) ;
    elif k <0 or k > n then
        0;
    else
        procname(n-1, k-1)+procname(n-1, k) ;
    end if;
end proc: # R. J. Mathar, Apr 26 2015

Mathematica

T[n_, k_] := T[n, k] = Which[k == 0 || k == n, 1, k == 1 || k == n-1, Floor[(3n-1)/2], k < 0 || k > n, 0, True, T[n-1, k-1] + T[n-1, k]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 30 2018 *)

Prog

(Haskell)
a026637 n k = a026637_tabl !! n !! k
a026637_row n = a026637_tabl !! n
a026637_tabl = [1] : [1, 1] : map (fst . snd)
   (iterate f (0, ([1, 2, 1], [0, 1, 1, 0]))) where
   f (i, (xs, ws)) = (1 - i,
     if i == 1 then (ys, ws) else (zipWith (+) ys ws, ws'))
        where ys = zipWith (+) ([0] ++ xs) (xs ++ [0])
              ws' = [0, 1, 0, 0] ++ drop 2 ws
-- Reinhard Zumkeller, Aug 08 2013

Crossrefs

Sequence in context: A176388 A282494 A156609 * A026659 A026386 A147532

Adjacent sequences: A026634 A026635 A026636 * A026638 A026639 A026640

Keyword

nonn,tabl,easy

Author

Clark Kimberling

Status

approved