A199881 Triangle T(n,k), read by rows, given by (1,-1,0,0,0,0,0,0,0,0,0,...) DELTA (1,0,-1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

1, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 3, 1, 0, 0, 1, 4, 4, 1, 0, 0, 0, 3, 7, 5, 1, 0, 0, 0, 1, 7, 11, 6, 1, 0, 0, 0, 0, 4, 14, 16, 7, 1, 0, 0, 0, 0, 1, 11, 25, 22, 8, 1, 0, 0, 0, 0, 0, 5, 25, 41, 29, 9, 1, 0, 0, 0, 0, 0, 1, 16, 50, 63, 37, 10, 1

Offset: 0

Author

Philippe Deléham, Nov 11 2011

Keywords

• nonn
• tabl

Comments

The nonzero entries of column k give row k+1 in A072405.

Examples

			Triangle begins:
  1;
  1, 1;
  0, 1, 1;
  0, 1, 2, 1; (key row for starting the recurrence)
  0, 0, 2, 3, 1;
  0, 0, 1, 4, 4,  1;
  0, 0, 0, 3, 7,  5,  1;
  0, 0, 0, 1, 7, 11,  6, 1;
  0, 0, 0, 0, 4, 14, 16, 7, 1;
		

Links

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

Crossrefs

Cf. A000931 (diagonal sums), A042950 (column sums), A055389 (row sums).

Cf. A000045, A028310, A072405, A084938, A104402.

Programs

Mathematica

T[n_, k_]:= T[n, k]= If[n<2, 1, If[k==0, 0, If[k==n, 1, If[n==2 && k==1, 1, T[n-1, k-1] +T[n-2, k-1] ]]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 28 2021 *)

Sage

def T(n,k): return binomial(k, n-k) + binomial(k+1, n-k-1)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 28 2021

Formula

T(n,k) = T(n-1,k-1) + T(n-2,k-1) starting with T(0,0) = T(1,0) = T(1,1) = T(2,1) = T(2,2) = 1 and T(2,0) = 0.

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

T(2n,n) = A028310(n).

From G. C. Greubel, Apr 28 2021: (Start)

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

T(n, k) = (-1)^(n-k)*A104402(n, k). (End)

From G. C. Greubel, Apr 30 2021: (Start)

Sum_{k=0..n} T(n, k) = 2*Fibonacci(n) + [n=0].

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