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

1,3

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

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.

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)

Example

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)

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 *)

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
(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

Crossrefs

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

Sequence in context: A145118 A124927 A126279 * A027144 A158303 A304223

Adjacent sequences: A135834 A135835 A135836 * A135838 A135839 A135840

Keyword

nice,nonn,tabl

Author

Gary W. Adamson, Dec 01 2007

Status

approved