A124137 A signed aerated and skewed version of A038137.

1, 0, 1, -1, 0, 2, 0, -2, 0, 3, 1, 0, -5, 0, 5, 0, 3, 0, -10, 0, 8, -1, 0, 9, 0, -20, 0, 13, 0, -4, 0, 22, 0, -38, 0, 21, 1, 0, -14, 0, 51, 0, -71, 0, 34, 0, 5, 0, -40, 0, 111, 0, -130, 0, 55, -1, 0, 20, 0, -105, 0, 233, 0, -235, 0, 89

Offset

0,6

Links

G. C. Greubel, Rows n=0..100 of triangle, flattened

Formula

T(n,k) = T(n-1,k-1) + T(n-2,k-2) - T(n-2,k), T(0,0)=1, T(n,k)=0 if k<0 or if n<k . T(n,n) = Fibonacci(n+1) = A000045(n+1).

Sum_{0<=k<=n} x^k*T(n,k)= A014983(n+1), A033999(n), A056594(n), A000012(n), A015518(n+1), A015525(n+1) for x=-2, -1, 0, 1, 2, 3 respectively.

Example

Triangle begins:
1;
0, 1;
-1, 0, 2;
0, -2, 0, 3;
1, 0, -5, 0, 5;
0, 3, 0, -10, 0, 8;
-1, 0, 9, 0, -20, 0, 13;
0, -4, 0, 22, 0, -38, 0, 21;
1, 0, -14, 0, 51, 0, -71, 0, 34;
0, 5, 0, -40, 0, 111, 0, -130, 0, 55;

Mathematica

T[0, 0]:= 1; T[n_, n_]:= Fibonacci[n + 1]; T[n_, k_]:= T[n, k] = If[k < 0 || n < k, 0, T[n - 1, k - 1] + T[n - 2, k - 2] - T[n - 2, k]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten  (* G. C. Greubel, May 27 2018 *)

Prog

(PARI) {T(n, k) = if(n==0 && k==0, 1, if(k==n, fibonacci(n+1), if(k<0 || n<k, 0, T(n-1, k-1) + T(n-2, k-2) - T(n-2, k))))};
for(n=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, May 27 2018

Crossrefs

Cf. A038137, A208336, A000045, A001629, A001628, A001872, A001873

Sequence in context: A261079 A182485 A164960 * A164092 A302643 A319973

Adjacent sequences: A124134 A124135 A124136 * A124138 A124139 A124140

Keyword

sign,tabl

Author

Philippe Deléham, Nov 30 2006

Extensions

Corrected and extended by Philippe Deléham, Apr 05 2012

Status

approved