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A236076 A skewed version of triangular array A122075.

%I A236076 #18 May 21 2019 03:48:37
%S A236076 1,0,2,0,1,3,0,0,3,5,0,0,1,7,8,0,0,0,4,15,13,0,0,0,1,12,30,21,0,0,0,0,
%T A236076 5,31,58,34,0,0,0,0,1,18,73,109,55,0,0,0,0,0,6,54,162,201,89,0,0,0,0,
%U A236076 0,1,25,145,344,365,144,0,0,0,0,0,0,7,85,361
%N A236076 A skewed version of triangular array A122075.
%C A236076 Triangle T(n,k), 0 <= k <= n, read by rows, given by (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
%C A236076 Subtriangle of the triangle A122950.
%H A236076 Reinhard Zumkeller, <a href="/A236076/b236076.txt">Rows n = 0..125 of triangle, flattened</a>
%H A236076 H. Fuks and J.M.G. Soto, <a href="http://arxiv.org/abs/1306.1189">Exponential convergence to equilibrium in cellular automata asymptotically emulating identity</a>, arXiv:1306.1189 [nlin.CG], 2013.
%F A236076 G.f.: (1+x*y)/(1 - x*y - x^2*y - x^2*y^2).
%F A236076 T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2), T(0,0)=1, T(1,0) = 0, T(1,1) = 2, T(n,k) = 0 if k < 0 or if k > n.
%F A236076 Sum_{k=0..n} T(n,k) = 2^n = A000079(n).
%F A236076 Sum_{n>=k} T(n,k) = A078057(k) = A001333(k+1).
%F A236076 T(n,n) = Fibonacci(n+2) = A000045(n+2).
%F A236076 T(n+1,n) = A023610(n-1), n >= 1.
%F A236076 T(n+2,n) = A129707(n).
%e A236076 Triangle begins:
%e A236076   1;
%e A236076   0,  2;
%e A236076   0,  1,  3;
%e A236076   0,  0,  3,  5;
%e A236076   0,  0,  1,  7,  8;
%e A236076   0,  0,  0,  4, 15, 13;
%e A236076   0,  0,  0,  1, 12, 30, 21;
%e A236076   0,  0,  0,  0,  5, 31, 58, 34;
%t A236076 T[n_, k_]:= If[k<0 || k>n, 0, If[n==0 && k==0, 1, If[k==0, 0, If[n==1 && k==1, 2, T[n-1, k-1] + T[n-2, k-1] + T[n-2, k-2]]]]]; Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 21 2019 *)
%o A236076 (Haskell)
%o A236076 a236076 n k = a236076_tabl !! n !! k
%o A236076 a236076_row n = a236076_tabl !! n
%o A236076 a236076_tabl = [1] : [0, 2] : f [1] [0, 2] where
%o A236076    f us vs = ws : f vs ws where
%o A236076      ws = [0] ++ zipWith (+) (zipWith (+) ([0] ++ us) (us ++ [0])) vs
%o A236076 -- _Reinhard Zumkeller_, Jan 19 2014
%o A236076 (PARI)
%o A236076 {T(n,k) = if(k<0 || k>n, 0, if(n==0 && k==0, 1, if(k==0, 0, if(n==1 && k==1, 2, T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2) ))))}; \\ _G. C. Greubel_, May 21 2019
%o A236076 (Sage)
%o A236076 def T(n, k):
%o A236076     if (k<0 or k>n): return 0
%o A236076     elif (n==0 and k==0): return 1
%o A236076     elif (k==0): return 0
%o A236076     elif (n==1 and k==1): return 2
%o A236076     else: return T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2)
%o A236076 [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, May 21 2019
%Y A236076 Cf. variant: A055830, A122075, A122950, A208337.
%Y A236076 Cf. A167704 (diagonal sums), A000079 (row sums).
%Y A236076 Cf. A111006.
%K A236076 easy,nonn,tabl
%O A236076 0,3
%A A236076 _Philippe Deléham_, Jan 19 2014