A080247 Formal inverse of triangle A080246. Unsigned version of A080245.

1, 2, 1, 6, 4, 1, 22, 16, 6, 1, 90, 68, 30, 8, 1, 394, 304, 146, 48, 10, 1, 1806, 1412, 714, 264, 70, 12, 1, 8558, 6752, 3534, 1408, 430, 96, 14, 1, 41586, 33028, 17718, 7432, 2490, 652, 126, 16, 1, 206098

Offset

0,2

Comments

Row sums are little Schroeder numbers A001003. Diagonal sums are generalized Fibonacci numbers A006603. Columns include A006318, A006319, A006320, A006321.

T(n,k) is the number of dissections of a convex (n+3)-gon by nonintersecting diagonals with exactly k diagonals emanating from a fixed vertex. Example: T(2,1)=4 because the dissections of the convex pentagon ABCDE having exactly one diagonal emanating from the vertex A are: {AC}, {AD}, {AC,EC} and {AD,BD}. - Emeric Deutsch, May 31 2004

For more triangle sums, see A180662, see the Schroeder triangle A033877 which is the mirror of this triangle. - Johannes W. Meijer, Jul 15 2013

Links

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)

Paul Barry, Laurent Biorthogonal Polynomials and Riordan Arrays, arXiv preprint arXiv:1311.2292 [math.CA], 2013.

Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.

James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.

P. Flajolet and M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math. 204 (1999), 203-229.

Shishuo Fu and Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019.

W.-j. Woan, The Lagrange inversion formula and divisibility properties, JIS 10 (2007) 07.7.8, example 5.

Formula

G.f.: 2/(2+y*x-y+y*(x^2-6*x+1)^(1/2))/y/x. - Vladeta Jovovic, Feb 16 2003

Essentially same triangle as triangle T(n,k), n > 0 and k > 0, read by rows; given by [0, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...] DELTA A000007 where DELTA is Deléham's operator defined in A084938.

T(n, k) = T(n-1, k-1) + 2*Sum_{j>=0} T(n-1, k+j) with T(0, 0) = 1 and T(n, k)=0 if k < 0. - Philippe Deléham, Jan 19 2004

T(n, k) = (k+1)*Sum_{j=0..n-k} (binomial(n+1, k+j+1)*binomial(n+j, j))/(n+1). - Emeric Deutsch, May 31 2004

Recurrence: T(0,0)=1; T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n,k+1). - David Callan, Jul 03 2006

T(n, k) = binomial(n, k)*hypergeom([k - n, n + 1], [k + 2], -1). - Peter Luschny, Jan 08 2018

T(n,k) = (k+1)/(n+1)*Sum_{m=0..n-k} 2^m*binomial(n+1,m)*binomial(n-k-1,n-k-m). - Vladimir Kruchinin, Jan 10 2022

Example

Triangle starts:
[0]    1
[1]    2,    1
[2]    6,    4,   1
[3]   22,   16,   6,   1
[4]   90,   68,  30,   8,  1
[5]  394,  304, 146,  48, 10,  1
[6] 1806, 1412, 714, 264, 70, 12, 1
...
From Gary W. Adamson, Jul 25 2011: (Start)
n-th row = top row of M^n, M = the following infinite square production matrix:
  2, 1, 0, 0, 0, ...
  2, 2, 1, 0, 0, ...
  2, 2, 2, 1, 0, ...
  2, 2, 2, 2, 1, ...
  ... (End)

Maple

A080247:=(n, k)->(k+1)*add(binomial(n+1, k+j+1)*binomial(n+j, j), j=0..n-k)/(n+1):
seq(seq(A080247(n, k), k=0..n), n=0..9);

Mathematica

Clear[w] w[n_, k_] /; k < 0 || k > n := 0 w[0, 0]=1 ; w[n_, k_] /; 0 <= k <= n && !n == k == 0 := w[n, k] = w[n-1, k-1] + w[n-1, k] + w[n, k+1] Table[w[n, k], {n, 0, 10}, {k, 0, n}] (* David Callan, Jul 03 2006 *)
T[n_, k_] := Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, -1];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Peter Luschny, Jan 08 2018 *)

Prog

(Sage)
def A080247_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1, k-1)-2*sum(prec(n, k+i-1) for i in (2..n-k+1))
    return [(-1)^(n-k)*prec(n, k) for k in (1..n)]
for n in (1..10): print(A080247_row(n)) # Peter Luschny, Mar 16 2016
(Maxima)
T(n, k):=((k+1)*sum(2^m*binomial(n+1, m)*binomial(n-k-1, n-k-m), m, 0, n-k))/(n+1); /* Vladimir Kruchinin, Jan 10 2022 */

Crossrefs

Cf. A000007, A033877 (mirror), A084938.

Sequence in context: A110681 A117852 A080245 * A078937 A167560 A132159

Adjacent sequences: A080244 A080245 A080246 * A080248 A080249 A080250

Keyword

nonn,tabl

Author

Paul Barry, Feb 15 2003

Status

approved