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A127895 Riordan array (1/(1+x)^3, x/(1+x)^3).

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