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 A112466 Riordan array ((1+2x)/(1+x), x/(1+x)). 4

%I

%S 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 -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 -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

%N Riordan array ((1+2x)/(1+x), x/(1+x)).

%C Row sums are (1,2,0,0,0,...).

%C Inverse is A112465.

%C 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

%H Michael De Vlieger, <a href="/A112466/b112466.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened)

%H 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 E. 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 Number triangle T(n,k) = (-1)^(n-k)*(C(n, n-k) - 2*C(n-1, n-k-1)).

%F Sum_{k=0..floor(n/2)} T(n-k,k) = (-1)^(n+1)*Fibonacci(n-2).

%F T(2n,n) = 0.

%F Sum_{k=0..n} T(n,k)*x^k = (x+1)*(x-1)^(n-1), for n >= 1. - _Philippe Deléham_, Oct 03 2005

%F 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 G.f.: (1+2*x)/(1+x-x*y). - _R. J. Mathar_, Aug 11 2015

%e Triangle starts

%e 1;

%e 1, 1;

%e -1, 0, 1;

%e 1, -1, -1, 1;

%e -1, 2, 0, -2, 1;

%e 1, -3, 2, 2, -3, 1;

%e -1, 4, -5, 0, 5, -4, 1;

%e From _Paul Barry_, Apr 08 2011: (Start)

%e Production matrix begins

%e 1, 1;

%e -2, -1, 1;

%e 2, 0, -1, 1;

%e -2, 0, 0, -1, 1;

%e 2, 0, 0, 0, -1, 1;

%e -2, 0, 0, 0, 0, -1, 1;

%e 2, 0, 0, 0, 0, 0, -1, 1; (End)

%p 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 {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 (PARI) T(n,k) = (-1)^(n-k)*(binomial(n, n-k) - 2*binomial(n-1, n-k-1)); \\ _Michel Marcus_, Feb 19 2020

%Y Cf. A008482, A037012, A097808, A112467.

%K easy,sign,tabl

%O 0,12

%A _Paul Barry_, Sep 06 2005

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Last modified April 12 19:50 EDT 2021. Contains 342932 sequences. (Running on oeis4.)