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
KEYWORD
nonn,tabl
AUTHOR
Philippe Deléham, Jan 21 2009
STATUS
approved