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A344566 T(n, k) = (-1)^(n - k)*binomial(n - 1, k - 1)*hypergeom([-(n - k)/2, -(n - k - 1)/2], [1 - n], 4). Triangle read by rows, T(n, k) for 0 <= k <= n.

%I A344566 #7 May 24 2021 23:34:12
%S A344566 1,0,1,0,-1,1,0,0,-2,1,0,1,1,-3,1,0,-1,2,3,-4,1,0,0,-4,2,6,-5,1,0,1,2,
%T A344566 -9,0,10,-6,1,0,-1,3,9,-15,-5,15,-7,1,0,0,-6,3,24,-20,-14,21,-8,1,0,1,
%U A344566 3,-18,-6,49,-21,-28,28,-9,1
%N A344566 T(n, k) = (-1)^(n - k)*binomial(n - 1, k - 1)*hypergeom([-(n - k)/2, -(n - k - 1)/2], [1 - n], 4). Triangle read by rows, T(n, k) for 0 <= k <= n.
%C A344566 The inverse of the Riordan array for directed animals A122896. Without the first column (1, 0, 0, ...) the inverse of the Motzkin triangle A064189.
%F A344566 Riordan_array (1, x / (1 + x + x^2)).
%e A344566 Triangle starts:
%e A344566 [0] 1;
%e A344566 [1] 0,  1;
%e A344566 [2] 0, -1,  1;
%e A344566 [3] 0,  0, -2,  1;
%e A344566 [4] 0,  1,  1, -3,   1;
%e A344566 [5] 0, -1,  2,  3,  -4,   1;
%e A344566 [6] 0,  0, -4,  2,   6,  -5,   1;
%e A344566 [7] 0,  1,  2, -9,   0,  10,  -6, 1;
%e A344566 [8] 0, -1,  3,  9, -15,  -5,  15, -7,  1;
%e A344566 [9] 0,  0, -6,  3,  24, -20, -14, 21, -8, 1.
%p A344566 T := (n,k) -> (-1)^(n-k)*binomial(n-1,k-1)*hypergeom([-(n-k)/2, -(n-k-1)/2], [1-n], 4): seq(seq(simplify(T(n, k)), k=0..n), n = 0..10);
%o A344566 (SageMath) # uses[riordan_array from A256893]
%o A344566 riordan_array(1, x / (1 + x + x^2), 10)
%Y A344566 A117569 (row sums).
%Y A344566 Cf. A122896, A064189.
%K A344566 sign,tabl
%O A344566 0,9
%A A344566 _Peter Luschny_, May 23 2021