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A112466 Riordan array ((1+2*x)/(1+x), x/(1+x)).

%I A112466 #54 May 03 2025 10:19:47
%S A112466 1,1,1,-1,0,1,1,-1,-1,1,-1,2,0,-2,1,1,-3,2,2,-3,1,-1,4,-5,0,5,-4,1,1,
%T A112466 -5,9,-5,-5,9,-5,1,-1,6,-14,14,0,-14,14,-6,1,1,-7,20,-28,14,14,-28,20,
%U A112466 -7,1,-1,8,-27,48,-42,0,42,-48,27,-8,1,1,-9,35,-75,90,-42,-42,90,-75,35,-9,1,-1,10,-44,110,-165,132,0,-132,165,-110,44,-10,1
%N A112466 Riordan array ((1+2*x)/(1+x), x/(1+x)).
%C A112466 Inverse is A112465.
%C A112466 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Aug 07 2006; corrected by _Philippe Deléham_, Dec 11 2008
%C A112466 Equals A097808 when the first column is removed. - _Georg Fischer_, Jul 26 2023
%H A112466 Michael De Vlieger, <a href="/A112466/b112466.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened)
%H A112466 Paul Barry, <a href="https://arxiv.org/abs/1912.01124">A Note on Riordan Arrays with Catalan Halves</a>, arXiv:1912.01124 [math.CO], 2019.
%H A112466 Emeric Deutsch, L. Ferrari and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.aam.2004.05.002">Production Matrices</a>, Advances in Mathematics, 34 (2005) pp. 101-122.
%F A112466 Number triangle: T(n,k) = (-1)^(n-k)*(C(n, n-k) - 2*C(n-1, n-k-1)), with T(0,0) = 1.
%F A112466 T(2*n, n) = 0 (main diagonal).
%F A112466 Sum_{k=0..n} T(n, k) = 0 + [n=0] + 2*[n=1] (row sums).
%F A112466 Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^(n+1)*Fibonacci(n-2) (diagonal sums).
%F A112466 Sum_{k=0..n} T(n,k)*x^k = (x+1)*(x-1)^(n-1), for n >= 1. - _Philippe Deléham_, Oct 03 2005
%F A112466 T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if n < 0 or if n < k, T(n,k) = T(n-1,k-1) - T(n-1,k) for n > 1. - _Philippe Deléham_, Nov 26 2006
%F A112466 G.f.: (1+2*x)/(1+x-x*y). - _R. J. Mathar_, Aug 11 2015
%F A112466 From _G. C. Greubel_, Apr 30 2025: (Start)
%F A112466 T(2*n+1, 2*n+1-k) = T(2*n+1, k) (symmetric odd n rows).
%F A112466 T(2*n, 2*n-k) = (-1)*T(2*n, k) (antisymmetric even n rows).
%F A112466 Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n) (signed row sums).
%F A112466 Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (-1)^n*A057079(n+2) (signed diagonal sums). (End)
%e A112466 Triangle starts
%e A112466    1;
%e A112466    1,  1;
%e A112466   -1,  0,  1;
%e A112466    1, -1, -1,  1;
%e A112466   -1,  2,  0, -2,  1;
%e A112466    1, -3,  2,  2, -3,  1;
%e A112466   -1,  4, -5,  0,  5, -4,  1;
%e A112466 From _Paul Barry_, Apr 08 2011: (Start)
%e A112466 Production matrix begins
%e A112466    1,  1;
%e A112466   -2, -1,  1;
%e A112466    2,  0, -1,  1;
%e A112466   -2,  0,  0, -1,  1;
%e A112466    2,  0,  0,  0, -1,  1;
%e A112466   -2,  0,  0,  0,  0, -1,  1;
%e A112466    2,  0,  0,  0,  0,  0, -1,  1; (End)
%p A112466 seq(seq( (-1)^(n-k)*(2*binomial(n-1, k-1)-binomial(n, k)), k=0..n), n=0..10); # _G. C. Greubel_, Feb 19 2020
%t A112466 {1}~Join~Table[(Binomial[n, n - k] - 2 Binomial[n - 1, n - k - 1])*(-1)^(n - k), {n, 12}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Feb 18 2020 *)
%o A112466 (PARI) T(n,k) = (-1)^(n-k)*(binomial(n, n-k) - 2*binomial(n-1, n-k-1)); \\ _Michel Marcus_, Feb 19 2020
%o A112466 (Magma)
%o A112466 A112466:= func< n,k | n eq 0 select 1 else (-1)^(n+k)*(Binomial(n,k) - 2*Binomial(n-1,k)) >;
%o A112466 [A112466(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 30 2025
%o A112466 (SageMath)
%o A112466 def A112466(n,k): return 1 if (n==0) else (-1)^(n+k)*(binomial(n,k) - 2*binomial(n-1,k))
%o A112466 print(flatten([[A112466(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Apr 30 2025
%Y A112466 Cf. A000007, A008482, A037012, A057079, A097808, A112467.
%Y A112466 Columns: A248157(n+2) (k=1), (-1)^n*A080956(n-2) (k=2), (-1)^(n-1)*A254749(n-2) (k=3).
%K A112466 easy,sign,tabl
%O A112466 0,12
%A A112466 _Paul Barry_, Sep 06 2005