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

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, 51, 40, 20, 6, 1, 0, 21, 71, 111, 105, 65, 27, 7, 1, 0, 34, 130, 233, 256, 190, 98, 35, 8, 1, 0, 55, 235, 474, 594, 511, 315, 140, 44, 9, 1, 0, 89, 420, 942, 1324, 1295, 924, 490, 192, 54, 10, 1

Offset: 0

Author

Philippe Deléham, Jan 21 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins:
[0] 1;
[1] 0,  1;
[2] 0,  1,   1;
[3] 0,  2,   2,   1;
[4] 0,  3,   5,   3,   1;
[5] 0,  5,  10,   9,   4,   1;
[6] 0,  8,  20,  22,  14,   5,  1;
[7] 0, 13,  38,  51,  40,  20,  6,  1;
[8] 0, 21,  71, 111, 105,  65, 27,  7, 1;
[9] 0, 34, 130, 233, 256, 190, 98, 35, 8, 1.
		

Links

• Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
• Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.

Crossrefs

Row sums are in A215928.

Central terms: T(2*n,n) = A213684(n) for n > 0.

Cf. A000045, A037027, A122542, A059283, A321620.

Programs

Haskell

a155161 n k = a155161_tabl !! n !! k
a155161_row n = a155161_tabl !! n
a155161_tabl = [1] : [0,1] : f [0] [0,1] where
   f us vs = ws : f vs ws where
     ws = zipWith (+) (us ++ [0,0]) $ zipWith (+) ([0] ++ vs) (vs ++ [0])
-- Reinhard Zumkeller, Apr 17 2013

Maple

T := (n, k) -> binomial(n-1, k-1)*hypergeom([-(n-k)/2, -(n-k-1)/2], [1-n], -4):
seq(seq(simplify(T(n, k)), k = 0..n), n = 0..11); # Peter Luschny, May 23 2021
# Uses function PMatrix from A357368.
PMatrix(10, n -> combinat:-fibonacci(n)); # Peter Luschny, Oct 07 2022

Mathematica

CoefficientList[#, y]& /@ CoefficientList[(1-x-x^2)/(1-x-x^2-x*y)+O[x]^12, x] // Flatten (* Jean-François Alcover, Mar 01 2019 *)
(* Generates the triangle without the leading '1' (rows are rearranged). *)
(* Function RiordanSquare defined in A321620. *)
RiordanSquare[x/(1 - x - x^2), 11] // Flatten  (* Peter Luschny, Feb 27 2021 *)

Maxima

M(n,k):=pochhammer(n,k)/k!;
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 */

Formula

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.

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

Recurrence: a(n+2,k+1) = a(n+1,k+1) + a(n+1,k) + a(n,k+1). - Emanuele Munarini, Mar 15 2011

G.f.: (1-x-x^2)/(1-x-x^2-x*y). - Philippe Deléham, Feb 08 2012

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

T(n, k) = binomial(n-1, k-1)*hypergeom([-(n-k)/2, -(n-k-1)/2], [1-n], -4). - Peter Luschny, May 23 2021