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

0,8

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.

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

Example

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.

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 *)

Prog

(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 */
(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

Crossrefs

Row sums are in A215928.

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

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

Sequence in context: A285308 A276543 A107424 * A185937 A292086 A065177

Adjacent sequences: A155158 A155159 A155160 * A155162 A155163 A155164

Keyword

nonn,tabl

Author

Philippe Deléham, Jan 21 2009

Status

approved