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A051632 Rows of triangle formed using Pascal's rule except we begin and end the n-th row with n-2.

%I A051632 #15 Oct 27 2023 22:00:44
%S A051632 -2,-1,-1,0,-2,0,1,-2,-2,1,2,-1,-4,-1,2,3,1,-5,-5,1,3,4,4,-4,-10,-4,4,
%T A051632 4,5,8,0,-14,-14,0,8,5,6,13,8,-14,-28,-14,8,13,6,7,19,21,-6,-42,-42,
%U A051632 -6,21,19,7,8,26,40,15,-48,-84,-48,15,40,26,8,9,34,66,55,-33,-132,-132
%N A051632 Rows of triangle formed using Pascal's rule except we begin and end the n-th row with n-2.
%C A051632 Row sums are -2.
%H A051632 Reinhard Zumkeller, <a href="/A051632/b051632.txt">Rows n=0..100 of triangle, flattened</a>
%H A051632 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A051632 -t(n,k)=((2*k + 1 - n)/(k + 1))*Binomial[n, k] + ((1 - n + 2 (-k + n))/(1 - k + n)) Binomial[n, -k + n]. [From _Roger L. Bagula_, Feb 17 2009]
%e A051632 Contribution from Roger L. Bagula, Feb 17 2009: (Start)
%e A051632 The rows of the triangle, negated, are:
%e A051632 {2},
%e A051632 {1, 1},
%e A051632 {0, 2, 0},
%e A051632 {-1, 2, 2, -1},
%e A051632 {-2, 1, 4, 1, -2},
%e A051632 {-3, -1, 5,5, -1, -3},
%e A051632 {-4, -4, 4, 10, 4, -4, -4},
%e A051632 {-5, -8, 0, 14, 14, 0, -8, -5},
%e A051632 {-6, -13, -8, 14, 28, 14, -8, -13, -6},
%e A051632 {-7, -19, -21, 6, 42,42, 6, -21, -19, -7},
%e A051632 {-8, -26, -40, -15, 48, 84, 48, -15, -40, -26, -8} (End)
%t A051632 t[n_, k_] = ((2*k + 1 - n)/(k + 1))*Binomial[n, k] + ((1 - n + 2 (-k + n))/(1 - k + n)) Binomial[n, -k + n];
%t A051632 Table[Table[t[n, k], {k, 0, n}], {n, 0, 10}];
%t A051632 Flatten[%] (* Roger L. Bagula , Feb 17 2009 *)
%o A051632 (Haskell)
%o A051632 a051632 n k = a051632_tabl !! n !! k
%o A051632 a051632_list = concat a051632_tabl
%o A051632 a051632_tabl = iterate (\rs -> zipWith (+) ([1] ++ rs) (rs ++ [1])) [-2]
%o A051632 -- _Reinhard Zumkeller_, Aug 21 2011
%Y A051632 A156644 [From Roger L. Bagula, Feb 17 2009]
%Y A051632 Cf. A007318.
%K A051632 easy,nice,sign,tabl
%O A051632 0,1
%A A051632 _Asher Auel_