|
%I #23 May 01 2021 02:14:51
%S 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,
%T 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,
%U 9,1,0,0,0,0,0,1,16,50,63,37,10,1
%N 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.
%C The nonzero entries of column k give row k+1 in A072405.
%H G. C. Greubel, <a href="/A199881/b199881.txt">Rows n = 0..50 of the triangle, flattened</a>
%F 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.
%F G.f.: (1+x-y*x^2)/(1-y*x-y*x^2).
%F T(2n,n) = A028310(n).
%F From _G. C. Greubel_, Apr 28 2021: (Start)
%F T(n, k) = binomial(k, n-k) + binomial(k+1, n-k-1).
%F T(n, k) = (-1)^(n-k)*A104402(n, k). (End)
%F From _G. C. Greubel_, Apr 30 2021: (Start)
%F Sum_{k=0..n} T(n, k) = 2*Fibonacci(n) + [n=0].
%F Sum_{n=k..2*k+1} T(n,k) = 3*2^(n-1) + (1/2)*[n=0]. (End)
%e Triangle begins:
%e 1;
%e 1, 1;
%e 0, 1, 1;
%e 0, 1, 2, 1; (key row for starting the recurrence)
%e 0, 0, 2, 3, 1;
%e 0, 0, 1, 4, 4, 1;
%e 0, 0, 0, 3, 7, 5, 1;
%e 0, 0, 0, 1, 7, 11, 6, 1;
%e 0, 0, 0, 0, 4, 14, 16, 7, 1;
%t 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] ]]]];
%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Apr 28 2021 *)
%o (Sage)
%o def T(n,k): return binomial(k, n-k) + binomial(k+1, n-k-1)
%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 28 2021
%Y Cf. A000931 (diagonal sums), A042950 (column sums), A055389 (row sums).
%Y Cf. A000045, A028310, A072405, A084938, A104402.
%K nonn,tabl
%O 0,9
%A _Philippe Deléham_, Nov 11 2011
|
