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

0,12

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

Links

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

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)

Example

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;

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A194527 A104244 A116403 * A185158 A185700 A368494

Adjacent sequences: A123146 A123147 A123148 * A123150 A123151 A123152

Keyword

nonn,tabl,easy

Author

Philippe Deléham, Nov 05 2006

Status

approved