A026615 Triangle read by rows: T(n,0) = T(n,n) = 1 for n >= 0, T(n,1) = T(n,n-1) = 2*n-1 for n >= 1, T(n,k) = T(n-1,k-1) + T(n-1,k) for 2 <= k <= n-2 and n >= 4.

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 10, 7, 1, 1, 9, 17, 17, 9, 1, 1, 11, 26, 34, 26, 11, 1, 1, 13, 37, 60, 60, 37, 13, 1, 1, 15, 50, 97, 120, 97, 50, 15, 1, 1, 17, 65, 147, 217, 217, 147, 65, 17, 1, 1, 19, 82, 212, 364, 434, 364, 212, 82, 19, 1

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabl

Comments

T(n, k) is the 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) for i=0, j >= 0 and for j=0, i >= 0.

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  3,  1;
  1,  5,  5,   1;
  1,  7, 10,   7,   1;
  1,  9, 17,  17,   9,   1;
  1, 11, 26,  34,  26,  11,   1;
  1, 13, 37,  60,  60,  37,  13,   1;
  1, 15, 50,  97, 120,  97,  50,  15,  1;
  1, 17, 65, 147, 217, 217, 147,  65, 17,  1;
  1, 19, 82, 212, 364, 434, 364, 212, 82, 19,  1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A026616, A026617, A026618, A026619, A026620, A026621, A026623.

Cf. A026624, A026956, A026957, A026958, A026959, A026960.

Cf. A000012, A005408, A098749, A145066, A176742.

Cf. A026622 (row sums), A026625 (diagonal sums).

Programs

Magma

function T(n,k) // T = A026615
   if k eq 0 or k eq n then return 1;
   elif k eq 1 or k eq n-1 then return 2*n-1;
   else return T(n-1, k-1) + T(n-1, k);
   end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 13 2024

Mathematica

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k==1 || k==n-1, 2*n-1, T[n-1, k -1] + T[n-1,k]]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 13 2024 *)

SageMath

def T(n,k): # T = A026615
    if k==0 or k==n: return 1
    elif k==1 or k==n-1: return 2*n-1
    else: return T(n-1, k-1) + T(n-1, k)
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jun 13 2024

Formula

Sum_{k=0..n} T(n, k) = A026622(n) (row sums).

From G. C. Greubel, Jun 13 2024: (Start)

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

T(n, 0) = A000012(n).

T(n, 1) = A005408(n-1), n >= 1.

T(n, 2) = A098749(n), n >= 2.

T(n, 3) = A145066(n-2) - [n=3], n >= 3.

Sum_{k=0..n} (-1)^k*T(n, k) = A176742(n) + [n=2].

Sum_{k=0..n} (-1)^k*T(n-k, k) = b(n-2) + 2*[n=0] + [n=1], where b(n) = (1/6)*(-2*sqrt(3)*sin(Pi*n/3) + 2*sqrt(3)*sin(5*Pi*n/3) + 3*cos(Pi* n/2) + 3*cos(3*Pi*n/2) - 6).

Sum_{k=0..n} k*T(n, k) = n*(7*2^(n-3) - 1) + (1/4)*[n=1]. (End)

Extensions

Offset corrected by G. C. Greubel, Jun 13 2024