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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 21 14:57 EDT 2018. Contains 316424 sequences. (Running on oeis4.)