A124137 - OEIS

%I #14 May 27 2018 01:45:17

%S 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,

%T -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,

%U 55,-1,0,20,0,-105,0,233,0,-235,0,89

%N A signed aerated and skewed version of A038137.

%H G. C. Greubel, <a href="/A124137/b124137.txt">Rows n=0..100 of triangle, flattened</a>

%F 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).

%F 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.

%e Triangle begins:

%e 1;

%e 0, 1;

%e -1, 0, 2;

%e 0, -2, 0, 3;

%e 1, 0, -5, 0, 5;

%e 0, 3, 0, -10, 0, 8;

%e -1, 0, 9, 0, -20, 0, 13;

%e 0, -4, 0, 22, 0, -38, 0, 21;

%e 1, 0, -14, 0, 51, 0, -71, 0, 34;

%e 0, 5, 0, -40, 0, 111, 0, -130, 0, 55;

%t 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 *)

%o (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))))};

%o for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, May 27 2018

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

%K sign,tabl

%O 0,6

%A _Philippe Deléham_, Nov 30 2006

%E Corrected and extended by _Philippe Deléham_, Apr 05 2012