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A112465 Riordan array (1/(1+x),x/(1-x)). 5

%I

%S 1,-1,1,1,0,1,-1,1,1,1,1,0,2,2,1,-1,1,2,4,3,1,1,0,3,6,7,4,1,-1,1,3,9,

%T 13,11,5,1,1,0,4,12,22,24,16,6,1,-1,1,4,16,34,46,40,22,7,1,1,0,5,20,

%U 50,80,86,62,29,8,1,-1,1,5,25,70,130,166,148,91,37,9,1,1,0,6,30,95,200,296,314,239,128,46,10,1

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

%C Row sums are A078008. Diagonal sums are A078024. Inverse is A112466. Note that C(n,k) = sum{j = 0..n-k, C(j+k-1,j)}.

%C Central coefficients T(2n, n) are A072547. - _Paul Barry_, Apr 08 2011

%C T(n,k) = A108561(n, n-k). - _Reinhard Zumkeller_, Jan 03 2014

%H Reinhard Zumkeller, <a href="/A112465/b112465.txt">Rows n = 0..125 of triangle, flattened</a>

%H Roland Bacher, <a href="http://arxiv.org/abs/1509.09054">Chebyshev polynomials, quadratic surds and a variation of Pascal's triangle</a>, arXiv:1509.09054 [math.CO], 2015.

%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.

%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>

%F Number triangle T(n, k) = sum{j = 0..n-k, C(j+k-1, j)*(-1)^(n-k-j)}.

%F T(n, 0) = (-1)^n, T(n, n) = 1, T(n+1, k) = T(n, k-1) + T(n, k), 0 < k < n. - _Reinhard Zumkeller_, Jan 03 2014

%F T(n, k) = T(n-1, k-1) + T(n-2,k) + T(n-2,k-1), T(0, 0) = 1, T(1, 0) = -1, T(1, 1) = 1, T(n, k) = 0 if k < 0 or if k > n. - _Philippe Deléham_, Jan 11 2014

%F exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(-1 + x + x^2/2! + x^3/3!) = -1 + 2*x^2/2! + 6*x^3/3! + 13*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - _Peter Bala_, Dec 21 2014

%e Triangle starts

%e 1;

%e -1,1;

%e 1,0,1;

%e -1,1,1,1;

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

%e -1,1,2,4,3,1;

%e 1,0,3,6,7,4,1;

%e Production matrix begins

%e -1, 1,

%e 0, 1, 1,

%e 0, 0, 1, 1,

%e 0, 0, 0, 1, 1,

%e 0, 0, 0, 0, 1, 1,

%e 0, 0, 0, 0, 0, 1, 1,

%e 0, 0, 0, 0, 0, 0, 1, 1,

%e 0, 0, 0, 0, 0, 0, 0, 1, 1

%e - _Paul Barry_, Apr 08 2011

%t T[n_, k_] := Sum[Binomial[j + k - 1, j]*(-1)^(n - k - j), {j, 0, n - k}];

%t Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 23 2018 *)

%o (Haskell)

%o a112465 n k = a112465_tabl !! n !! k

%o a112465_row n = a112465_tabl !! n

%o a112465_tabl = iterate f [1] where

%o f xs'@(x:xs) = zipWith (+) ([-x] ++ xs ++ [0]) ([0] ++ xs')

%o -- _Reinhard Zumkeller_, Jan 03 2014

%Y Cf. A112468, A059260.

%K easy,sign,tabl

%O 0,13

%A _Paul Barry_, Sep 06 2005

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Last modified January 27 15:14 EST 2020. Contains 331295 sequences. (Running on oeis4.)