A124927 Triangle read by rows: T(n,0)=1, T(n,k)=2*binomial(n,k) if k>0 (0<=k<=n).

1, 1, 2, 1, 4, 2, 1, 6, 6, 2, 1, 8, 12, 8, 2, 1, 10, 20, 20, 10, 2, 1, 12, 30, 40, 30, 12, 2, 1, 14, 42, 70, 70, 42, 14, 2, 1, 16, 56, 112, 140, 112, 56, 16, 2, 1, 18, 72, 168, 252, 252, 168, 72, 18, 2, 1, 20, 90, 240, 420, 504, 420, 240, 90, 20, 2, 1, 22, 110, 330, 660, 924, 924, 660, 330, 110, 22, 2

Offset

0,3

Comments

Pascal triangle with all entries doubled except for the first entry in each row. A028326 with first column replaced by 1's. Row sums are 2^(n+1)-1.

From Paul Barry, Sep 19 2008: (Start)

Reversal of A129994. Diagonal sums are A001595. T(2n,n) is A100320.

Binomial transform of matrix with 1,2,2,2,... on main diagonal, zero elsewhere. (End)

This sequence is jointly generated with A210042 as an 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)+v(n-1,x) +1 and v(n,x)=x*u(n-1,x)+x*v(n-1,x). See the Mathematica section. - Clark Kimberling, Mar 09 2012

Subtriangle of the triangle given by (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 25 2012

Links

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened

Index entries for triangles and arrays related to Pascal's triangle

Formula

T(n,0) = 1; for n>0: T(n,n) = 2, T(n,k) = T(n-1,k) + T(n-1,n-k), 1<k<n. - Reinhard Zumkeller, Mar 04 2012

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,0) = 1, T(1,1) = 2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 25 2012

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

Example

Triangle starts:
  1;
  1,  2;
  1,  4,  2;
  1,  6,  6,  2;
  1,  8, 12,  8,  2;
  1, 10, 20, 20, 10, 2;
(1, 0, 0, 1, 0, 0, ...) DELTA (0, 2, -1, 0, 0, ...) begins:
  1;
  1,  0;
  1,  2,  0;
  1,  4,  2,  0;
  1,  6,  6,  2,  0;
  1,  8, 12,  8,  2, 0;
  1, 10, 20, 20, 10, 2, 0. - Philippe Deléham, Mar 25 2012

Maple

T:=proc(n, k) if k=0 then 1 else 2*binomial(n, k) fi end: for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form

Mathematica

(* First program *)
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
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[%]    (* A210042 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A124927 *) (* Clark Kimberling, Mar 17 2012 *)
(* Second program *)
Table[If[k==0, 1, 2*Binomial[n, k]], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 10 2019 *)

Prog

(Haskell)
a124927 n k = a124927_tabl !! n !! k
a124927_row n = a124927_tabl !! n
a124927_tabl = iterate
   (\row -> zipWith (+) ([0] ++ reverse row) (row ++ [1])) [1]
-- Reinhard Zumkeller, Mar 04 2012
(PARI) T(n, k) = if(k==0, 1, 2*binomial(n, k)); \\ G. C. Greubel, Jul 10 2019
(Magma) [k eq 0 select 1 else 2*Binomial(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 10 2019
(Sage)
def T(n, k):
    if (k==0): return 1
    else: return 2*binomial(n, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jul 10 2019

Crossrefs

Cf. A000225.

Cf. A074909.

Sequence in context: A133113 A124840 A145118 * A126279 A135837 A027144

Adjacent sequences: A124924 A124925 A124926 * A124928 A124929 A124930

Keyword

nonn,easy,tabl

Author

Gary W. Adamson, Nov 12 2006

Extensions

Edited by N. J. A. Sloane, Nov 24 2006

Status

approved