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

%I A176239 #7 Mar 30 2012 18:52:05
%S A176239 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,
%T A176239 -42,0,0,0,0,1,-5,14,-28,42,-42,0,132,0,0,0,0,0,1,-6,20,-48,90,-132,
%U A176239 132,0,-429,0,0,0,0,0,0,1,-7,27,-75,165,-297,429,-429,0,1430
%N A176239 Shifted signed Catalan triangle T(n,k) = (-1)^*(n+k+1)*A009766(n,k-n+1) read by rows.
%F A176239 T(n,k) = T(n+1,k)+T(n+1,k+1), k <= 2n+1.
%F A176239 T(n,2n) = 0.
%F A176239 T(n,2n+1) = (-1)^(n+1)*A000108(n+1).
%F A176239 T(n,k) =  (-1)^(n+k+1)*A009766(n,k-n+1), k < 2n.
%e A176239 The triangle starts in row n=0 with columns 0 <= k < 2*(n+1) as:
%e A176239 0,-1;                          (-1)^k*k  A001477
%e A176239 1,-1,.0,.2;                      (-1)^(k+1)*(k+1)*(k-2)/2  A080956, A000096
%e A176239 0,.1,-2,.2,.0,-5;                 (-1)^n*k*(k+1)*(k-4)/6 A129936, A005586
%e A176239 0,.0,.1,-3,.5,-5,..0,.14;             (-1)^k*k*(k+1)*(k-1)*(k-6)/24, A005587
%e A176239 0,.0,.0,.1,-4,.9,-14,.14,.0,-42;           A005557, A034807
%e A176239 0,.0,.0,.0,.1,-5,.14,-28,42,-42,0,132;
%p A176239 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:
%p A176239 A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:
%p A176239 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
%Y A176239 Cf A140344, A033184.
%K A176239 tabf,sign
%O A176239 0,6
%A A176239 _Paul Curtz_, Apr 12 2010