A026736 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 n is even and k=(n-2)/2, otherwise T(n,k) = T(n-1,k-1) + T(n-1,k).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 6, 4, 1, 1, 6, 11, 10, 5, 1, 1, 7, 22, 21, 15, 6, 1, 1, 8, 29, 43, 36, 21, 7, 1, 1, 9, 37, 94, 79, 57, 28, 8, 1, 1, 10, 46, 131, 173, 136, 85, 36, 9, 1, 1, 11, 56, 177, 398, 309, 221, 121, 45, 10, 1, 1, 12, 67, 233, 575, 707, 530, 342, 166, 55, 11, 1

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabl
• walk

Comments

T(n, k) is the 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, i+2)-to-(i+1, i+3) for i >= 0.

Examples

			Triangle begins
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,   1;
  1,  5,  6,   4,   1;
  1,  6, 11,  10,   5,   1;
  1,  7, 22,  21,  15,   6,   1;
  1,  8, 29,  43,  36,  21,   7,   1;
  1,  9, 37,  94,  79,  57,  28,   8,   1;
  1, 10, 46, 131, 173, 136,  85,  36,   9,   1;
  1, 11, 56, 177, 398, 309, 221, 121,  45,  10,   1;
  1, 12, 67, 233, 575, 707, 530, 342, 166,  55,  11,  1;
  ...
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Crossrefs

Row sums give A026743.

T(2n,n) gives A026737(n) or A111279(n+1).

Programs

GAP

T:= function(n,k)
    if k=0 or k=n then return 1;
    elif (n mod 2)=0 and k=Int((n-2)/2) then return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k);
    else return T(n-1, k-1) + T(n-1, k);
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Jul 16 2019

Mathematica

T[, 0] = T[n, n_] = 1; T[n_, k_] := T[n, k] = If[EvenQ[n] && k == (n-2)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 22 2018 *)

PARI

T(n,k) = if(k==n || k==0, 1, if((n%2)==0 && k==(n-2)/2, T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) ));
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jul 16 2019

Sage

def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k==(n-2)/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
    else: return T(n-1, k-1) + T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jul 16 2019

Extensions

Offset corrected by Alois P. Heinz, Jul 23 2018