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

0,5

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.

Links

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

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)

Example

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;

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 *)

Prog

(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
(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

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).

Sequence in context: A130154 A208328 A134398 * A026681 A109128 A113245

Adjacent sequences: A026612 A026613 A026614 * A026616 A026617 A026618

Keyword

nonn,tabl

Author

Clark Kimberling

Extensions

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

Status

approved