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

Author

Gary W. Adamson, Nov 12 2006

Keywords

• nonn
• easy
• tabl

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

Examples

			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
		

Links

• Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A000225.

Cf. A074909.

Programs

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

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

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 *)

PARI

T(n,k) = if(k==0,1, 2*binomial(n,k)); \\ 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

Formula

T(n,0) = 1; for n>0: T(n,n) = 2, T(n,k) = T(n-1,k) + T(n-1,n-k), 1Reinhard 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

Extensions

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