A026637 Triangular array T read by rows: T(n,0) = T(n,n) = 1 for n >= 0, T(n,1) = T(n,n-1) = floor((3*n-1)/2) for n >= 1, otherwise 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, 223, 439, 610, 610, 439, 223, 76, 16, 1

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabl
• easy

Comments

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.

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

Examples

			Triangle begins as:
  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;
		

Links

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

Crossrefs

Cf. A026638, A026639, A026640, A026641, A026642, A026643, A026644.

Cf. A026966, A026967, A026968, A026969, A026970, A228053.

Sums include: A000007 (alternating sign row), A026644 (row), A026645, A026646, A026647 (diagonal).

Programs

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

Magma

function T(n,k) // T = A026637
   if k eq 0 or k eq n then return 1;
   elif k eq 1 or k eq n-1 then return Floor((3*n-1)/2);
   else return T(n-1, k) + T(n-1, k-1);
   end if;
end function;
[T(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jun 28 2024

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 *)

SageMath

def T(n,k): # T = A026637
    if k==0 or k==n: return 1
    elif k==1 or k==n-1: return ((3*n-1)//2)
    else: return T(n-1, k) + T(n-1, k-1)
flatten([[T(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Jun 28 2024

Formula

From G. C. Greubel, Jun 28 2024: (Start)

T(n, n-k) = T(n, k).

T(2*n-1, n-1) = A026641(n), n >= 1.

Sum_{k=0..n} T(n, k) = A026644(n).

Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n). (End)