The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A155161 A Fibonacci convolution triangle: Riordan array (1, x/(1 - x - x^2)). Triangle T(n,k), 0 <= k <= n, read by rows. 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 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 21 09:22 EDT 2021. Contains 345358 sequences. (Running on oeis4.)