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A155161 Triangle T(n,k), 0<=k<=n, read by rows given by [0,1,1,-1,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938 . 15
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

A Fibonacci convolution triangle ; Riordan array (1,x/(1-x-x^2)) .

Row sums are in A215928 . - Philippe Deléham, Aug 31 2012

Central terms: T(2*n,n) = A213684(n) for n > 0. - Reinhard Zumkeller, Apr 17 2013

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

a(n,k) = sum(M(k,i)*binomial(i,n-i-k),i,0,n-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<=k<=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

EXAMPLE

Triangle begins :

1 ;

0,1 ;

0,1,1 ;

0,2,2,1 ;

0,3,5,3,1 ;

0,5,10,9,4,1 ;

...

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 *)

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

Cf. A000045, A037027.

Cf. A122542, A059283.

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

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Last modified December 12 20:12 EST 2019. Contains 329961 sequences. (Running on oeis4.)