A152815 Triangle T(n,k), read by rows given by [1,0,-1,0,0,0,0,0,0,...] DELTA [0,1,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 2, 1, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 1, 4, 6, 4, 1, 0, 0, 0, 0, 1, 4, 6, 4, 1, 0, 0, 0, 0, 0, 1, 5, 10, 10, 5, 1, 0, 0, 0, 0, 0, 1, 5, 10, 10, 5, 1, 0, 0, 0, 0, 0, 0, 1, 6, 15, 20, 15, 6, 1, 0, 0, 0, 0, 0, 0, 1, 6, 15, 20, 15, 6, 1, 0, 0, 0

Offset: 0

Author

Philippe Deléham, Dec 13 2008

Keywords

• easy
• nonn
• tabl

Comments

Triangle read by rows, Pascal's triangle (A007318) rows repeated.

Riordan array (1/(1-x), x^2/(1-x^2)). - Philippe Deléham, Feb 27 2012

Examples

			Triangle begins:
  1;
  1, 0;
  1, 1, 0;
  1, 1, 0, 0;
  1, 2, 1, 0, 0;
  1, 2, 1, 0, 0, 0;
  1, 3, 3, 1, 0, 0, 0;
  1, 3, 3, 1, 0, 0, 0, 0;
  1, 4, 6, 4, 1, 0, 0, 0, 0; ...
		

Links

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

Crossrefs

Cf. A007318, A064861, A152198 (another version), A000931 (diagonal sums), A016116 (row sums).

Programs

Haskell

a152815 n k = a152815_tabl !! n !! k
a152815_row n = a152815_tabl !! n
a152815_tabl = [1] : [1,0] : t [1,0] where
   t ys = zs : zs' : t zs' where
     zs' = zs ++ [0]; zs = zipWith (+) ([0] ++ ys) (ys ++ [0])
-- Reinhard Zumkeller, Feb 28 2012

Mathematica

m = 13;
(* DELTA is defined in A084938 *)
DELTA[Join[{1, 0, -1}, Table[0, {m}]], Join[{0, 1, -1}, Table[0, {m}]], m] // Flatten (* Jean-François Alcover, Feb 19 2020 *)
T[n_, k_] := If[n<0, 0, Binomial[Floor[n/2], k]]; (* Michael Somos, Oct 01 2022 *)

PARI

{T(n, k) = if(n<0, 0, binomial(n\2, k))}; /* Michael Somos, Oct 01 2022 */

Formula

T(n,k) = T(n-1,k) + ((1+(-1)^n)/2)*T(n-1,k-1).

G.f.: (1+x)/(1-(1+y)*x^2).

Sum_{k=0..n} T(n,k)*x^k = A000012(n), A016116(n), A108411(n), A213173(n), A074872(n+1) for x = 0,1,2,3,4 respectively. - Philippe Deléham, Nov 26 2011, Apr 22 2013

Extensions

Example corrected by Philippe Deléham, Dec 13 2008