A176239 Shifted signed Catalan triangle T(n,k) = (-1)^*(n+k+1)*A009766(n,k-n+1) read by rows.

0, -1, 1, -1, 0, 2, 0, 1, -2, 2, 0, -5, 0, 0, 1, -3, 5, -5, 0, 14, 0, 0, 0, 1, -4, 9, -14, 14, 0, -42, 0, 0, 0, 0, 1, -5, 14, -28, 42, -42, 0, 132, 0, 0, 0, 0, 0, 1, -6, 20, -48, 90, -132, 132, 0, -429, 0, 0, 0, 0, 0, 0, 1, -7, 27, -75, 165, -297, 429, -429, 0, 1430

Offset: 0

Author

Paul Curtz, Apr 12 2010

Keywords

• tabf
• sign

Examples

			The triangle starts in row n=0 with columns 0 <= k < 2*(n+1) as:
0,-1;                          (-1)^k*k  A001477
1,-1,.0,.2;                      (-1)^(k+1)*(k+1)*(k-2)/2  A080956, A000096
0,.1,-2,.2,.0,-5;                 (-1)^n*k*(k+1)*(k-4)/6 A129936, A005586
0,.0,.1,-3,.5,-5,..0,.14;             (-1)^k*k*(k+1)*(k-1)*(k-6)/24, A005587
0,.0,.0,.1,-4,.9,-14,.14,.0,-42;           A005557, A034807
0,.0,.0,.0,.1,-5,.14,-28,42,-42,0,132;
		

Crossrefs

Cf A140344, A033184.

Programs

Maple

A009766 := proc(n,k) if k<0 or k >n then 0; else binomial(n+k,n)*(n-k+1)/(n+1) ; end if; end proc:
A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:
A176239 := proc(n,k) if k <= 2*n-1 then (-1)^(n+k+1)*A009766(n,k-n+1) elif k = 2*n then 0; elif k < 2*(n+1) then (-1)^(n+1)*A000108(n+1); else 0; end if; end proc: # R. J. Mathar, Dec 03 2010

Formula

T(n,k) = T(n+1,k)+T(n+1,k+1), k <= 2n+1.

T(n,2n) = 0.

T(n,2n+1) = (-1)^(n+1)*A000108(n+1).

T(n,k) = (-1)^(n+k+1)*A009766(n,k-n+1), k < 2n.