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%I #46 Oct 07 2022 07:23:00
%S 1,0,1,0,1,1,0,2,2,1,0,3,5,3,1,0,5,10,9,4,1,0,8,20,22,14,5,1,0,13,38,
%T 51,40,20,6,1,0,21,71,111,105,65,27,7,1,0,34,130,233,256,190,98,35,8,
%U 1,0,55,235,474,594,511,315,140,44,9,1,0,89,420,942,1324,1295,924,490,192,54,10,1
%N A Fibonacci convolution triangle: Riordan array (1, x/(1 - x - x^2)). Triangle T(n,k), 0 <= k <= n, read by rows.
%H Reinhard Zumkeller, <a href="/A155161/b155161.txt">Rows n = 0..120 of triangle, flattened</a>
%H Paul Barry, <a href="http://dx.doi.org/10.1016/j.laa.2015.10.032">Riordan arrays, generalized Narayana triangles, and series reversion</a>, Linear Algebra and its Applications, 491 (2016) 343-385.
%F T(n, k) given by [0,1,1,-1,0,0,0,...] DELTA [1,0,0,0,...] where DELTA is the operator defined in A084938.
%F a(n,k) = Sum_{i=0..n-k} M(k,i)*binomial(i,n-i-k), where M(n,k) = n(n+1)(n+2)...(n+k-1)/k!. - _Emanuele Munarini_, Mar 15 2011
%F Recurrence: a(n+2,k+1) = a(n+1,k+1) + a(n+1,k) + a(n,k+1). - _Emanuele Munarini_, Mar 15 2011
%F G.f.: (1-x-x^2)/(1-x-x^2-x*y). - _Philippe Deléham_, Feb 08 2012
%F Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000129(n) (n > 0), A052991(n), A155179(n), A155181(n), A155195(n), A155196(n), A155197(n), A155198(n), A155199(n) for x = 0,1,2,3,4,5,6,7,8,9 respectively. - _Philippe Deléham_, Feb 08 2012
%F T(n, k) = binomial(n-1, k-1)*hypergeom([-(n-k)/2, -(n-k-1)/2], [1-n], -4). - _Peter Luschny_, May 23 2021
%e Triangle begins:
%e [0] 1;
%e [1] 0, 1;
%e [2] 0, 1, 1;
%e [3] 0, 2, 2, 1;
%e [4] 0, 3, 5, 3, 1;
%e [5] 0, 5, 10, 9, 4, 1;
%e [6] 0, 8, 20, 22, 14, 5, 1;
%e [7] 0, 13, 38, 51, 40, 20, 6, 1;
%e [8] 0, 21, 71, 111, 105, 65, 27, 7, 1;
%e [9] 0, 34, 130, 233, 256, 190, 98, 35, 8, 1.
%p T := (n, k) -> binomial(n-1, k-1)*hypergeom([-(n-k)/2, -(n-k-1)/2], [1-n], -4):
%p seq(seq(simplify(T(n, k)), k = 0..n), n = 0..11); # _Peter Luschny_, May 23 2021
%p # Uses function PMatrix from A357368.
%p PMatrix(10, n -> combinat:-fibonacci(n)); # _Peter Luschny_, Oct 07 2022
%t CoefficientList[#, y]& /@ CoefficientList[(1-x-x^2)/(1-x-x^2-x*y)+O[x]^12, x] // Flatten (* _Jean-François Alcover_, Mar 01 2019 *)
%t (* Generates the triangle without the leading '1' (rows are rearranged). *)
%t (* Function RiordanSquare defined in A321620. *)
%t RiordanSquare[x/(1 - x - x^2), 11] // Flatten (* _Peter Luschny_, Feb 27 2021 *)
%o (Maxima) M(n,k):=pochhammer(n,k)/k!;
%o create_list(sum(M(k,i)*binomial(i,n-i-k),i,0,n-k),n,0,8,k,0,n); /* _Emanuele Munarini_, Mar 15 2011 */
%o (Haskell)
%o a155161 n k = a155161_tabl !! n !! k
%o a155161_row n = a155161_tabl !! n
%o a155161_tabl = [1] : [0,1] : f [0] [0,1] where
%o f us vs = ws : f vs ws where
%o ws = zipWith (+) (us ++ [0,0]) $ zipWith (+) ([0] ++ vs) (vs ++ [0])
%o -- _Reinhard Zumkeller_, Apr 17 2013
%Y Row sums are in A215928.
%Y Central terms: T(2*n,n) = A213684(n) for n > 0.
%Y Cf. A000045, A037027, A122542, A059283, A321620.
%K nonn,tabl
%O 0,8
%A _Philippe Deléham_, Jan 21 2009
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