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A176239
Shifted signed Catalan triangle T(n,k) = (-1)^*(n+k+1)*A009766(n,k-n+1) read by rows.
4
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
0,6
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.
EXAMPLE
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;
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
CROSSREFS
KEYWORD
tabf,sign
AUTHOR
Paul Curtz, Apr 12 2010
STATUS
approved