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 A135837 A007318 * a triangle with (1, 2, 2, 4, 4, 8, 8,...) in the main diagonal and the rest zeros. 5
 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, 14, 42, 140, 140, 168, 56, 16, 1, 16, 56, 224, 280, 448, 224, 128, 16, 1, 18, 72, 336, 504, 1008, 672, 576, 144, 32 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Row sums = A001333 starting (1, 3, 7, 17, 41, 99, 239,...). A135837 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)=2x*u(n-1,x)+v(n-1,x).  See the Mathematica section.  [From Clark Kimberling, Feb 26 2012] 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 LINKS Reinhard Zumkeller, Rows n=1..150 of triangle, flattened FORMULA Binomial transform of a triangle with (1, 2, 2, 4, 4, 8, 8,...) in the main diagonal and the rest zeros. Contribution from Philippe Deléham, Mar 19 2012. (Start) As DELTA-triangle with 0<=k<=n : 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). 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) g.f.: -(1-x+2*x*y)*x*y/(-1+2*x+2*x^2*y^2-x^2). - R. J. Mathar, Aug 11 2015 EXAMPLE First few rows of the triangle are: 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, 0, 0, 1, 0, 0, ...) DELTA (0, 2, -1, -1, 0, 0, ...) begins : 1 1, 0 1, 2, 0 1, 4, 2, 0 1, 6, 6, 4, 0 1, 8, 12, 16, 4, 0 1, 10, 20, 40, 20, 8, 0 1, 12, 30, 80, 60, 48, 8, 0. Philippe Deléham, Mar 19 2012 MATHEMATICA u[1, x_] := 1; v[1, x_] := 1; z = 13; u[n_, x_] := u[n - 1, x] + x*v[n - 1, x]; v[n_, x_] := 2 x*u[n - 1, x] + v[n - 1, x]; Table[Expand[u[n, x]], {n, 1, z/2}] Table[Expand[v[n, x]], {n, 1, z/2}] cu = Table[CoefficientList[u[n, x], x], {n, 1, z}]; TableForm[cu] Flatten[%]    (* A117919 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%]    (* A135837 *) PROG (Haskell) a135837 n k = a135837_tabl !! (n-1) !! (k-1) a135837_row n = a135837_tabl !! (n-1) a135837_tabl = [1] : [1, 2] : f [1] [1, 2] where    f xs ys = ys' : f ys ys' where      ys' = zipWith3 (\u v w -> 2 * u - v + 2 * w)                     (ys ++ [0]) (xs ++ [0, 0]) ([0, 0] ++ xs) -- Reinhard Zumkeller, Aug 08 2012 CROSSREFS Cf. A001333, A135838, A117919. Sequence in context: A145118 A124927 A126279 * A027144 A158303 A304223 Adjacent sequences:  A135834 A135835 A135836 * A135838 A135839 A135840 KEYWORD nice,nonn AUTHOR Gary W. Adamson, Dec 01 2007 STATUS approved

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Last modified October 16 16:08 EDT 2019. Contains 328101 sequences. (Running on oeis4.)