OFFSET
0,5
COMMENTS
Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 2, 1, 2, 1, 2, 1, ...] DELTA [1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938 . Inverse is Riordan array (1, x*(1-x)/(1+x)).
T(n, r) gives the number of [0,r]-covering hierarchies with n segments terminating at r (see Kreweras work). - Michel Marcus, Nov 22 2014
LINKS
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973), see page 15.
FORMULA
T(n,0) = 0^n, T(n,k) = T(n-1,k-1)+T(n-1,k)+T(n,k+1) if k>0.
EXAMPLE
Triangle begins:
1;
0, 1:
0, 2, 1;
0, 6, 4, 1;
0, 22, 16, 6, 1;
0, 90, 68, 30, 8, 1;
Production matrix is:
0...1
0...2...1
0...2...2...1
0...2...2...2...1
0...2...2...2...2...1
0...2...2...2...2...2...1
0...2...2...2...2...2...2...1
0...2...2...2...2...2...2...2...1
0...2...2...2...2...2...2...2...2...1
... - Philippe Deléham, Feb 09 2014
MATHEMATICA
T[n_, n_] = 1; T[_, 0] = 0; T[n_, k_] := T[n, k] = T[n-1, k-1] + T[n-1, k] + T[n, k+1];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)
PROG
(Sage)
def A122538_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 (0..n)]
for n in (0..10): print(A122538_row(n)) # Peter Luschny, Mar 16 2016
CROSSREFS
Cf. A001003 (row sums).
KEYWORD
nonn,tabl
AUTHOR
Philippe Deléham, Sep 18 2006
STATUS
approved