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%I #20 Oct 09 2022 14:59:19
%S 1,-3,1,6,-6,1,-10,21,-9,1,15,-56,45,-12,1,-21,126,-165,78,-15,1,28,
%T -252,495,-364,120,-18,1,-36,462,-1287,1365,-680,171,-21,1,45,-792,
%U 3003,-4368,3060,-1140,231,-24,1,-55,1287,-6435,12376,-11628,5985,-1771,300,-27,1
%N Riordan array (1/(1+x)^3, x/(1+x)^3).
%C The matrix inverse of the convolution triangle of A001764 (number of ternary trees). - _Peter Luschny_, Oct 09 2022
%H G. C. Greubel, <a href="/A127895/b127895.txt">Rows n=0..100 of triangle, flattened</a>
%F T(n, k) = (-1)^(n-k)*binomial(n +2*k +2, n-k).
%F Sum_{k=0..n} T(n, k) = A127896(n) (row sums).
%F Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n*A095263(n) (diagonal sums).
%e Triangle begins
%e 1;
%e -3, 1;
%e 6, -6, 1;
%e -10, 21, -9, 1;
%e 15, -56, 45, -12, 1;
%e -21, 126, -165, 78, -15, 1;
%e 28, -252, 495, -364, 120, -18, 1;
%e -36, 462, -1287, 1365, -680, 171, -21, 1;
%e 45, -792, 3003, -4368, 3060, -1140, 231, -24, 1;
%e -55, 1287, -6435, 12376, -11628, 5985, -1771, 300, -27, 1;
%e 66, -2002, 12870, -31824, 38760, -26334, 10626, -2600, 378, -30, 1;
%p # Uses function InvPMatrix from A357585. Adds column 1, 0, 0, ... to the left.
%p InvPMatrix(10, n -> binomial(3*n, n)/(2*n+1)); # _Peter Luschny_, Oct 09 2022
%t Table[(-1)^(n-k)*Binomial[n+2*k+2, n-k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Apr 29 2018 *)
%o (PARI) for(n=0, 10, for(k=0,n, print1((-1)^(n-k)*binomial(n+2*k+2, n-k), ", "))) \\ _G. C. Greubel_, Apr 29 2018
%o (Magma) [(-1)^(n-k)*Binomial(n+2*k+2, n-k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Apr 29 2018
%o (Sage) flatten([[(-1)^(n-k)*binomial(n+2*k+2, n-k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 16 2021
%Y Inverse is A127898.
%Y Alternating sign version of A127893.
%Y Cf. A001764, A095263, A127896.
%K easy,sign,tabl
%O 0,2
%A _Paul Barry_, Feb 04 2007
%E Terms a(50) onward added by _G. C. Greubel_, Apr 29 2018
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