A135837 A007318 * a triangle with (1, 2, 2, 4, 4, 8, 8, ...) in the main diagonal and the rest zeros.

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

Offset: 1

Author

Gary W. Adamson, Dec 01 2007

Keywords

• nice
• nonn
• tabl

Comments

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

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

Examples

			First few rows of the triangle:
  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;
  ...
From _Philippe Deléham_, Mar 19 2012: (Start)
(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; (End)
		

Links

• Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened

Crossrefs

Cf. A000217, A001333 (row sums), A007318, A016116, A117919, A135838.

Programs

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

Mathematica

(* First program *)
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 *) (* Clark Kimberling, Feb 26 2012 *)
(* Second program *)
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]]]];
Table[T[n, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Feb 07 2022 *)

Sage

def T(n,k): # A135837
    if (k<1 or k>n): return 0
    elif (k==1): return 1
    elif (k==n): return 2^(n//2)
    else: return 2*T(n-1, k) - T(n-2, k) + 2*T(n-2, k-2)
flatten([[T(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 07 2022

Formula

Binomial transform of a triangle with (1, 2, 2, 4, 4, 8, 8, ...) in the main diagonal and the rest zeros.

Sum_{k=1..n} T(n, k) = A001333(n).

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.: x*y*(1-x+2*x*y)/(1-2*x-2*x^2*y^2+x^2). - R. J. Mathar, Aug 11 2015

From G. C. Greubel, Feb 07 2022: (Start)

T(n, n) = A016116(n).

T(n, 2) = 2*(n-1).

T(n, 3) = 2*A000217(n-2). (End)