A123149 Triangle T(n,k), 0<=k<=n, read by rows given by [1, 0, -1, 0, 0, 0, 0, 0, ...] DELTA [0, 1, 0, -1, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 3, 5, 5, 3, 1, 0, 1, 3, 6, 7, 6, 3, 1, 0, 1, 4, 9, 13, 13, 9, 4, 1, 0, 1, 4, 10, 16, 19, 16, 10, 4, 1, 0, 1, 5, 14, 26, 35, 35, 26, 14, 5, 1, 0, 1, 5, 15, 30, 45, 51, 45, 30, 15, 5, 1, 0, 1, 6, 20, 45, 75, 96, 96, 75, 45, 20, 6, 1, 0

Offset: 0

Author

Philippe Deléham, Nov 05 2006

Keywords

• nonn
• tabl
• easy

Comments

A169623 is a very similar triangle except it does not have the outer diagonal of 0's. - N. J. A. Sloane, Nov 23 2017

Examples

			Triangle begins:
  1;
  1,  0;
  1,  1,  0;
  1,  1,  1,  0;
  1,  2,  2,  1,  0;
  1,  2,  3,  2,  1,  0;
  1,  3,  5,  5,  3,  1,  0;
  1,  3,  6,  7,  6,  3,  1,  0;
  1,  4,  9, 13, 13,  9,  4,  1,  0;
		

Links

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

Crossrefs

Cf. A002426, A005773, A013979, A027907, A038754, A084938, A135528, A169623, A182522 (row sums).

Programs

Magma

function T(n,k) // T = A123149
  if k lt 0 or k gt n then return 0;
  elif k eq 0 or k eq n-1 then return 1;
  elif k eq n then return 0;
  else return T(n-2,k) +T(n-2,k-1) +T(n-2,k-2);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 17 2023

Mathematica

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0 || k==n-1, 1, If[k==n, 0, T[n-2,k] +T[n-2,k-1] +T[n-2,k-2] ]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 17 2023 *)

SageMath

def T(n,k): # T = A123149
    if (k<0 or k>n): return 0
    elif (k==0 or k==n-1): return 1
    elif (k==n): return 0
    else: return T(n-2,k) +T(n-2,k-1) +T(n-2,k-2)
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 17 2023

Formula

T(n,k) = T(n-1,k-1) + T(n-1,k) if n even, T(n,k) = T(n-1,k-1) + T(n-2,k) if n odd, T(0,0) = 1, T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k < 0 or if k > n.

T(n,k) = T(n,n-k-1).

Sum_{k=0..n} T(n,k) = A038754(n-1), for n>=1.

T(2*n,n) = A005773(n).

T(2*n+1,n) = A002426(n).

From Philippe Deléham, May 04 2012: (Start)

G.f.: (1+x-y^2*x^2)/(1-x^2-y*x^2-y^2*x^2).

T(n,k) = T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n.

Sum_{k=0..n} T(n,k) = A182522(n). (End)

From G. C. Greubel, Jul 17 2023: (Start)

Sum_{k=0..n} (-1)^k*T(n,k) = A135528(n).

Sum_{k=0..floor(n/2)} T(n-k,k) = [n==0] + A013979(n+1). (End)