|
| |
|
|
A080247
|
|
Formal inverse of triangle A080246. Unsigned version of A080245.
|
|
8
|
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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, 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, 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
|
|
|
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
|
|
|
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
|
| |
|
|
