A106579 Triangular array associated with Schroeder numbers: T(0,0) = 1, T(n,0) = 0 for n > 0; T(n,k) = 0 if k < n; T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k).

1, 0, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 6, 16, 22, 0, 1, 8, 30, 68, 90, 0, 1, 10, 48, 146, 304, 394, 0, 1, 12, 70, 264, 714, 1412, 1806, 0, 1, 14, 96, 430, 1408, 3534, 6752, 8558, 0, 1, 16, 126, 652, 2490, 7432, 17718, 33028, 41586, 0, 1, 18, 160, 938, 4080, 14002, 39152, 89898, 164512, 206098

Offset

0,6

Links

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973).

G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)

Formula

G.f.: Sum T(n, k)*x^n*y^k = 1 + y*(1 - x*y - (x^2*y^2 - 6*x*y + 1)^(1/2))/(2*y + x*y - 1 + (x^2*y^2 - 6*x*y + 1)^(1/2)).

Example

Triangle starts
  1;
  0,    1;
  0,    1,    2;
  0,    1,    4,    6;
  0,    1,    6,   16,   22;
  0,    1,    8,   30,   68,   90;
  0,    1,   10,   48,  146,  304,  394;
  0,    1,   12,   70,  264,  714, 1412, 1806;
  ...

Mathematica

T[n_, k_]:= T[n, k]= Which[n==k==0, 1, n==0, 0, k==0, 0, k>n, 0, True, T[n, k-1] + T[n-1, k-1] + T[n-1, k]]; Table[T[n, k], {n, 0, 11}, {k, 0, n}]//Flatten (* Michael De Vlieger, Nov 05 2017 *)

Prog

(Haskell)
a106579 n k = a106579_tabl !! n !! k
a106579_row n = a106579_tabl !! n
a106579_tabl = [1] : iterate
   (\row -> scanl1 (+) $ zipWith (+) ([0] ++ row) (row ++ [0])) [0, 1]
-- Reinhard Zumkeller, Apr 17 2013
(Sage)
def A106579_row(n):
    if n==0: return [1]
    @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)^k*prec(n, n-k+1) for k in (0..n)]
for n in (0..10): print(A106579_row(n)) # Peter Luschny, Mar 16 2016

Crossrefs

Essentially the same as A033877 except with a leading column 1, 0, 0, 0, ...

Last diagonal: A006318 or A103137.

Row sums give A001003.

See A033877 for more comments and references.

Sequence in context: A359107 A229223 A128749 * A378318 A287318 A329020

Adjacent sequences: A106576 A106577 A106578 * A106580 A106581 A106582

Keyword

nonn,tabl

Author

N. J. A. Sloane, May 30 2005

Status

approved