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
KEYWORD
AUTHOR
Gary W. Adamson, Dec 01 2007
STATUS
approved