A123149 - OEIS

%I #20 Jul 17 2023 17:14:23

%S 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,

%T 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,

%U 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

%N 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.

%C A169623 is a very similar triangle except it does not have the outer diagonal of 0's. - _N. J. A. Sloane_, Nov 23 2017

%H G. C. Greubel, <a href="/A123149/b123149.txt">Rows n = 0..50 of the triangle, flattened</a>

%F 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.

%F T(n,k) = T(n,n-k-1).

%F Sum_{k=0..n} T(n,k) = A038754(n-1), for n>=1.

%F T(2*n,n) = A005773(n).

%F T(2*n+1,n) = A002426(n).

%F From _Philippe Deléham_, May 04 2012: (Start)

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

%F 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.

%F Sum_{k=0..n} T(n,k) = A182522(n). (End)

%F From _G. C. Greubel_, Jul 17 2023: (Start)

%F Sum_{k=0..n} (-1)^k*T(n,k) = A135528(n).

%F Sum_{k=0..floor(n/2)} T(n-k,k) = [n==0] + A013979(n+1). (End)

%e Triangle begins:

%e   1;

%e   1,  0;

%e   1,  1,  0;

%e   1,  1,  1,  0;

%e   1,  2,  2,  1,  0;

%e   1,  2,  3,  2,  1,  0;

%e   1,  3,  5,  5,  3,  1,  0;

%e   1,  3,  6,  7,  6,  3,  1,  0;

%e   1,  4,  9, 13, 13,  9,  4,  1,  0;

%t 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] ]]];

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 17 2023 *)

%o (Magma)

%o function T(n,k) // T = A123149

%o   if k lt 0 or k gt n then return 0;

%o   elif k eq 0 or k eq n-1 then return 1;

%o   elif k eq n then return 0;

%o   else return T(n-2,k) +T(n-2,k-1) +T(n-2,k-2);

%o   end if;

%o end function;

%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 17 2023

%o (SageMath)

%o def T(n,k): # T = A123149

%o     if (k<0 or k>n): return 0

%o     elif (k==0 or k==n-1): return 1

%o     elif (k==n): return 0

%o     else: return T(n-2,k) +T(n-2,k-1) +T(n-2,k-2)

%o flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jul 17 2023

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

%K nonn,tabl,easy

%O 0,12

%A _Philippe Deléham_, Nov 05 2006