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)
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
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
KEYWORD
AUTHOR
Gary W. Adamson, Nov 12 2006
EXTENSIONS
Edited by N. J. A. Sloane, Nov 24 2006
STATUS
approved