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A155161 A Fibonacci convolution triangle: Riordan array (1, x/(1 - x - x^2)). Triangle T(n,k), 0 <= k <= n, read by rows.

%I A155161 #46 Oct 07 2022 07:23:00
%S A155161 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 A155161 51,40,20,6,1,0,21,71,111,105,65,27,7,1,0,34,130,233,256,190,98,35,8,
%U A155161 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 A155161 A Fibonacci convolution triangle: Riordan array (1, x/(1 - x - x^2)). Triangle T(n,k), 0 <= k <= n, read by rows.
%H A155161 Reinhard Zumkeller, <a href="/A155161/b155161.txt">Rows n = 0..120 of triangle, flattened</a>
%H A155161 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 A155161 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 A155161 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 A155161 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 A155161 G.f.: (1-x-x^2)/(1-x-x^2-x*y). - _Philippe Deléham_, Feb 08 2012
%F A155161 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 A155161 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 A155161 Triangle begins:
%e A155161 [0] 1;
%e A155161 [1] 0,  1;
%e A155161 [2] 0,  1,   1;
%e A155161 [3] 0,  2,   2,   1;
%e A155161 [4] 0,  3,   5,   3,   1;
%e A155161 [5] 0,  5,  10,   9,   4,   1;
%e A155161 [6] 0,  8,  20,  22,  14,   5,  1;
%e A155161 [7] 0, 13,  38,  51,  40,  20,  6,  1;
%e A155161 [8] 0, 21,  71, 111, 105,  65, 27,  7, 1;
%e A155161 [9] 0, 34, 130, 233, 256, 190, 98, 35, 8, 1.
%p A155161 T := (n, k) -> binomial(n-1, k-1)*hypergeom([-(n-k)/2, -(n-k-1)/2], [1-n], -4):
%p A155161 seq(seq(simplify(T(n, k)), k = 0..n), n = 0..11); # _Peter Luschny_, May 23 2021
%p A155161 # Uses function PMatrix from A357368.
%p A155161 PMatrix(10, n -> combinat:-fibonacci(n)); # _Peter Luschny_, Oct 07 2022
%t A155161 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 A155161 (* Generates the triangle without the leading '1' (rows are rearranged). *)
%t A155161 (* Function RiordanSquare defined in A321620. *)
%t A155161 RiordanSquare[x/(1 - x - x^2), 11] // Flatten  (* _Peter Luschny_, Feb 27 2021 *)
%o A155161 (Maxima) M(n,k):=pochhammer(n,k)/k!;
%o A155161 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 A155161 (Haskell)
%o A155161 a155161 n k = a155161_tabl !! n !! k
%o A155161 a155161_row n = a155161_tabl !! n
%o A155161 a155161_tabl = [1] : [0,1] : f [0] [0,1] where
%o A155161    f us vs = ws : f vs ws where
%o A155161      ws = zipWith (+) (us ++ [0,0]) $ zipWith (+) ([0] ++ vs) (vs ++ [0])
%o A155161 -- _Reinhard Zumkeller_, Apr 17 2013
%Y A155161 Row sums are in A215928.
%Y A155161 Central terms: T(2*n,n) = A213684(n) for n > 0.
%Y A155161 Cf. A000045, A037027, A122542, A059283, A321620.
%K A155161 nonn,tabl
%O A155161 0,8
%A A155161 _Philippe Deléham_, Jan 21 2009