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A122538 Riordan array (1, x*f(x)) where f(x)is the g.f. of A006318. 1
1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 22, 16, 6, 1, 0, 90, 68, 30, 8, 1, 0, 394, 304, 146, 48, 10, 1, 0, 1806, 1412, 714, 264, 70, 12, 1, 0, 8558, 6752, 3534, 1408, 430, 96, 14, 1, 0, 41586, 33028, 17718, 7432, 2490, 652, 126, 16, 1 (list; table; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..54.

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

Another version : A080247, A080245, A033877.

Columns: A000007, A006318, A006319, A006320, A006321.

Diagonals: A000012, A005843, A054000.

Cf. A001003 (row sums).

Sequence in context: A147720 A205813 A127631 * A090238 A047922 A276891

Adjacent sequences:  A122535 A122536 A122537 * A122539 A122540 A122541

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Sep 18 2006

STATUS

approved

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Last modified November 14 17:22 EST 2019. Contains 329126 sequences. (Running on oeis4.)