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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_{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

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 April 8 14:52 EDT 2020. Contains 333314 sequences. (Running on oeis4.)