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%I #29 Feb 07 2022 02:31:14
%S 1,1,2,1,4,2,1,6,6,4,1,8,12,16,4,1,10,20,40,20,8,1,12,30,80,60,48,8,1,
%T 14,42,140,140,168,56,16,1,16,56,224,280,448,224,128,16,1,18,72,336,
%U 504,1008,672,576,144,32
%N A007318 * a triangle with (1, 2, 2, 4, 4, 8, 8, ...) in the main diagonal and the rest zeros.
%C This sequence is jointly generated with A117919 as a triangular array of coefficients of polynomials v(n,x): initially, u(1,x) = v(1,x) = 1; for n > 1, u(n,x) = u(n-1,x) + x*v(n-1)x and v(n,x) = 2*x*u(n-1,x) + v(n-1,x). See the Mathematica section. - _Clark Kimberling_, Feb 26 2012
%C Subtriangle of the triangle (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 19 2012
%H Reinhard Zumkeller, <a href="/A135837/b135837.txt">Rows n = 1..150 of triangle, flattened</a>
%F Binomial transform of a triangle with (1, 2, 2, 4, 4, 8, 8, ...) in the main diagonal and the rest zeros.
%F Sum_{k=1..n} T(n, k) = A001333(n).
%F From _Philippe Deléham_, Mar 19 2012: (Start)
%F As DELTA-triangle with 0 <= k <= n:
%F G.f.: (1-x+2*y*x^2-2*y^2*x^2)/(1-2*x+2*y*x^2-2*y^2*x^2).
%F T(n,k) = 2*T(n-1,k) - T(n-2,k) + 2*T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 2, T(n,k) = 0 if k < 0 or if k > n. (End)
%F G.f.: x*y*(1-x+2*x*y)/(1-2*x-2*x^2*y^2+x^2). - _R. J. Mathar_, Aug 11 2015
%F From _G. C. Greubel_, Feb 07 2022: (Start)
%F T(n, n) = A016116(n).
%F T(n, 2) = 2*(n-1).
%F T(n, 3) = 2*A000217(n-2). (End)
%e First few rows of the triangle:
%e 1;
%e 1, 2;
%e 1, 4, 2;
%e 1, 6, 6, 4;
%e 1, 8, 12, 16, 4;
%e 1, 10, 20, 40, 20, 8;
%e 1, 12, 30, 80, 60, 48, 8;
%e ...
%e From _Philippe Deléham_, Mar 19 2012: (Start)
%e (1, 0, 0, 1, 0, 0, ...) DELTA (0, 2, -1, -1, 0, 0, ...) begins:
%e 1;
%e 1, 0;
%e 1, 2, 0;
%e 1, 4, 2, 0;
%e 1, 6, 6, 4, 0;
%e 1, 8, 12, 16, 4, 0;
%e 1, 10, 20, 40, 20, 8, 0;
%e 1, 12, 30, 80, 60, 48, 8, 0; (End)
%t (* First program *)
%t u[1, x_]:= 1; v[1, x_]:= 1; z = 13;
%t u[n_, x_]:= u[n-1, x] + x*v[n-1, x];
%t v[n_, x_]:= 2 x*u[n-1, x] + v[n-1, x];
%t Table[Expand[u[n, x]], {n, 1, z/2}]
%t Table[Expand[v[n, x]], {n, 1, z/2}]
%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t TableForm[cu]
%t Flatten[%] (* A117919 *)
%t Table[Expand[v[n, x]], {n, 1, z}]
%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t TableForm[cv]
%t Flatten[%] (* A135837 *) (* _Clark Kimberling_, Feb 26 2012 *)
%t (* Second program *)
%t T[n_, k_]:= T[n, k]= If[k<1 || k>n, 0, If[k==1, 1, If[k==n, 2^Floor[n/2], 2*T[n-1, k] - T[n-2, k] + 2*T[n-2, k-2]]]];
%t Table[T[n, k], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Feb 07 2022 *)
%o (Haskell)
%o a135837 n k = a135837_tabl !! (n-1) !! (k-1)
%o a135837_row n = a135837_tabl !! (n-1)
%o a135837_tabl = [1] : [1, 2] : f [1] [1, 2] where
%o f xs ys = ys' : f ys ys' where
%o ys' = zipWith3 (\u v w -> 2 * u - v + 2 * w)
%o (ys ++ [0]) (xs ++ [0, 0]) ([0, 0] ++ xs)
%o -- _Reinhard Zumkeller_, Aug 08 2012
%o (Sage)
%o def T(n,k): # A135837
%o if (k<1 or k>n): return 0
%o elif (k==1): return 1
%o elif (k==n): return 2^(n//2)
%o else: return 2*T(n-1, k) - T(n-2, k) + 2*T(n-2, k-2)
%o flatten([[T(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Feb 07 2022
%Y Cf. A000217, A001333 (row sums), A007318, A016116, A117919, A135838.
%K nice,nonn,tabl
%O 1,3
%A _Gary W. Adamson_, Dec 01 2007
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