A122538 Riordan array (1, x*f(x)) where f(x)is the g.f. of A006318.

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, 0, 206098, 164512, 89898, 39152, 14002, 4080, 938, 160, 18, 1

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. C. Greubel, Rows n = 0..100 of the triangle, flattened

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,k) = T(n-1,k-1) + T(n-1,k) + T(n,k+1) if k > 0, with T(n, 0) = 0^n, and T(n, n) = 1.

Sum_{k=0..n} T(n, k) = A001003(n).

From G. C. Greubel, Oct 27 2024: (Start)

T(2*n, n) = A103885(n).

Sum_{k=0..n} (-1)^k*T(n, k) = -A001003(n-1).

Sum_{k=0..floor(n/2)} T(n-k, k) = [n=0] + 0*[n=1] + A006603(n-2)*[n>1]. (End)

Example

Triangle begins:
  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;
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, 12}, {k, 0, n}]//Flatten (* 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..12): print(A122538_row(n)) # Peter Luschny, Mar 16 2016
(Magma)
function T(n, k) // T = A122538
  if k eq 0 then return 0^n;
  elif k eq n then return 1;
  else return T(n-1, k-1) + T(n-1, k) + T(n, k+1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 27 2024

Crossrefs

Another version : A080247, A080245, A033877.

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

Diagonals: A000012, A005843, A054000.

Sums include: A001003 (row and alternating sign), A006603 (diagonal).

Cf. A103885.

Sequence in context: A147720 A205813 A127631 * A090238 A358694 A047922

Adjacent sequences: A122535 A122536 A122537 * A122539 A122540 A122541

Keyword

nonn,tabl

Author

Philippe Deléham, Sep 18 2006

Status

approved