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

0,9

Comments

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

Links

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

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)

Example

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;

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 *)

Prog

(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

Crossrefs

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

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

Sequence in context: A120730 A122851 A064301 * A060701 A363916 A275345

Adjacent sequences: A199878 A199879 A199880 * A199882 A199883 A199884

Keyword

nonn,tabl

Author

Philippe Deléham, Nov 11 2011

Status

approved