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A106270 Inverse of number triangle A106268; triangle T(n,k), 0 <= k <= n.

%I A106270 #48 Jan 10 2023 08:00:08
%S A106270 1,-1,1,-2,-1,1,-5,-2,-1,1,-14,-5,-2,-1,1,-42,-14,-5,-2,-1,1,-132,-42,
%T A106270 -14,-5,-2,-1,1,-429,-132,-42,-14,-5,-2,-1,1,-1430,-429,-132,-42,-14,
%U A106270 -5,-2,-1,1,-4862,-1430,-429,-132,-42,-14,-5,-2,-1,1,-16796,-4862,-1430,-429,-132,-42,-14,-5,-2,-1,1
%N A106270 Inverse of number triangle A106268; triangle T(n,k), 0 <= k <= n.
%C A106270 Sequence array for the sequence a(n) = 2*0^n - C(n), where C = A000108 (Catalan numbers). Row sums are A106271. Antidiagonal sums are A106272.
%C A106270 The lower triangular matrix |T| (unsigned case) gives the Riordan matrix R = (c(x), x), a Toeplitz matrix. It is its own so called L-Eigen-matrix (cf. Bernstein - Sloane for such Eigen-sequences, and Barry for such eigentriangles), that is R*R = L*(R - I), with the infinite matrices I (identity) and L with matrix elements L(i, j) = delta(i,j-1) (Kronecker symbol; first upper diagonal with 1s). Thus R = L*(I - R^{-1}), and R^{-1} = I - L^{tr}*R (tr for transposed) is the Riordan matrix (1 - x*c(x), x) given in A343233. (For finite N X N matrices the R^{-1} equation is also valid, but for the other two ones the last row with only zeros has to be omitted.) - _Gary W. Adamson_ and _Wolfdieter Lang_, Apr 11 2021
%H A106270 Michel Marcus, <a href="/A106270/b106270.txt">Table of n, a(n) for n = 0..1325</a> (Rows n = 0..50 of triangle, flattened).
%H A106270 Paul Barry, <a href="http://arxiv.org/abs/1107.5490">Invariant number triangles, eigentriangles and Somos-4 sequences</a>, arXiv:1107.5490 [math.CO], 2011.
%H A106270 M. Bernstein and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0205301">Some canonical sequences of integers</a>, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%F A106270 Number triangle T(n, k) = 2*0^(n-k) - C(n-k) if k <= n, 0 otherwise; Riordan array (2*sqrt(1-4*x)/(1+sqrt(1-4*x)), x) = (c(x)*sqrt(1-4*x), x), where c(x) is the g.f. of A000108.
%F A106270 Sum_{k=0..n} T(n, k) = A106271(n).
%F A106270 Sum_{k=0..floor(n/2)} T(n, k) = A106272(n).
%F A106270 Bivariate g.f.: Sum_{n, k >= 0} T(n,k)*x^n*y^k = (1/(1 - x*y)) * (2 - c(x)), where c(x) is the g.f. of A000108. - _Petros Hadjicostas_, Jul 15 2019
%F A106270 From _G. C. Greubel_, Jan 09 2023: (Start)
%F A106270 Sum_{k=0..n} 2^(n-j)*abs(T(n,k)) = A112696(n).
%F A106270 Sum_{k=0..n} 2^k*abs(T(n,k)) = A014318(n). (End)
%e A106270 Triangle (with rows n >= 0 and columns k >= 0) begins as follows:
%e A106270      1;
%e A106270     -1,    1;
%e A106270     -2,   -1,   1;
%e A106270     -5,   -2,  -1,   1;
%e A106270    -14,   -5,  -2,  -1,  1;
%e A106270    -42,  -14,  -5,  -2, -1,  1;
%e A106270   -132,  -42, -14,  -5, -2, -1,  1;
%e A106270   -429, -132, -42, -14, -5, -2, -1,  1;
%t A106270 A106270[n_, k_]:= If[k==n, 1, -CatalanNumber[n-k]];
%t A106270 Table[A106270[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 09 2023 *)
%o A106270 (PARI) C(n) = binomial(2*n,n)/(n+1); \\ A000108
%o A106270 T(n, k) = if(k <= n, 2*0^(n-k) - C(n-k), 0); \\ _Michel Marcus_, Nov 11 2022
%o A106270 (Magma)
%o A106270 A106270:= func< n,k | k eq n select 1 else -Catalan(n-k) >;
%o A106270 [A106270(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 09 2023
%o A106270 (SageMath)
%o A106270 def A106270(n,k): return 1 if (k==n) else -catalan_number(n-k)
%o A106270 flatten([[A106270(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jan 09 2023
%Y A106270 Cf. A000108, A014318, A106268, A106271, A106272, A112696, A343233.
%K A106270 easy,sign,tabl
%O A106270 0,4
%A A106270 _Paul Barry_, Apr 28 2005