A033877 Triangular array read by rows associated with Schroeder numbers: T(1,k) = 1; T(n,k) = 0 if k < n; T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k).

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• nice

Comments

A106579 is in some ways a better version of this sequence, but since this was entered first it will be the main entry for this triangle.

The diagonals of this triangle are self-convolutions of the main diagonal A006318: 1, 2, 6, 22, 90, 394, 1806, ... . - Philippe Deléham, May 15 2005

From Johannes W. Meijer, Sep 22 2010, Jul 15 2013: (Start)

Note that for the terms T(n,k) of this triangle n indicates the column and k the row.

The triangle sums, see A180662, link Schroeder's triangle with several sequences, see the crossrefs. The mirror of this triangle is A080247.

Quite surprisingly the Kn1p sums, p >= 1, are all related to A026003 and crystal ball sequences for n-dimensional cubic lattices (triangle offset is 0): Kn11(n) = A026003(n), Kn12(n) = A026003(n+2) - 1, Kn13(n) = A026003(n+4) - A005408(n+3), Kn14(n) = A026003(n+6) - A001844(n+4), Kn15(n) = A026003(n+8) - A001845(n+5), Kn16(n) = A026003(n+10) - A001846(n+6), Kn17(n) = A026003(n+12) - A001847(n+7), Kn18(n) = A026003(n+14) - A001848(n+8), Kn19(n) = A026003(n+16) - A001849(n+9), Kn110(n) = A026003(n+18) - A008417(n+10), Kn111(n) = A026003(n+20) - A008419(n+11), Kn112(n) = A026003(n+22) - A008421(n+12). (End)

T(n,k) is the number of normal semistandard Young tableaux with two columns, one of height k and one of height n. The recursion can be seen by performing jeu de taquin deletion on all instances of the smallest value. (If there are two instances of the smallest value, jeu de taquin deletion will always shorten the right column first and the left column second.) - Jacob Post, Jun 19 2018

Examples

			Triangle starts:
  1;
  1,    2;
  1,    4,    6;
  1,    6,   16,   22;
  1,    8,   30,   68,   90;
  1,   10,   48,  146,  304,  394;
  1,   12,   70,  264,  714, 1412, 1806;
  ... - _Joerg Arndt_, Sep 29 2013
		

Links

• T. D. Noe, Rows k = 1..50 of triangle, flattened
• Henry Bottomley, Illustration of initial terms
• Kevin Brown, Hipparchus on Compound Statements, 1994-2010. - _Johannes W. Meijer_, Sep 22 2010
• James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
• G. Kreweras, Sur les hiérarchies de segments, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 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)
• G. Kreweras, Aires des chemins surdiagonaux et application à un problème économique, Cahiers du Bureau universitaire de recherche opérationnelle Série Recherche 24 (1976): 1-8. [Annotated scanned copy]
• J. W. Meijer, Famous numbers on a chessboard, Acta Nova, Volume 4, No.4, December 2010. pp. 589-598.
• J. M. Oh, An explicit formula for the number of fuzzy subgroups of a finite abelian p-group of rank two, Iranian Journal of Fuzzy Systems, Dec 2013, Vol. 10 Issue 6, pp. 125-135.
• E. Pergola and R. A. Sulanke, Schroeder Triangles, Paths and Parallelogram Polyominoes, J. Integer Sequences, 1 (1998), #98.1.7.
• S. Samieinia, The number of continuous curves in digital geometry, Port. Math. 67 (1) (2010) 75-89, last table.
• R. A. Sulanke, Objects counted by the central Delannoy numbers, J. Integer Seq. 6 (2003), Article 03.1.5, 19 pp.
• Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.

Crossrefs

Essentially same triangle as A080247 and A080245 but with rows read in reversed order. Also essentially the same triangle as A106579.

Cf. A000007, A006318, A006319, A006320, A006321, A008288, A080243.

Cf. A084938, A103885, A238112, A330801.

Cf. A001003 (row sums), A026003 (antidiagonal sums).

Triangle sums (see the comments): A001003 (Row1, Row2), A026003 (Kn1p, p >= 1), A006603 (Kn21), A227504 (Kn22), A227505 (Kn23), A006603(2*n) (Kn3), A001850 (Kn4), A227506 (Fi1), A010683 (Fi2).

Programs

Haskell

a033877 n k = a033877_tabl !! n !! k
a033877_row n = a033877_tabl !! n
a033877_tabl = iterate
   (\row -> scanl1 (+) $ zipWith (+) ([0] ++ row) (row ++ [0])) [1]
-- Reinhard Zumkeller, Apr 17 2013

Magma

function t(n,k)
  if k le 0 or k gt n then return 0;
  elif k eq 1 then return 1;
  else return t(n,k-1) + t(n-1,k-1) + t(n-1,k);
  end if;
end function;
[t(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 23 2023

Maple

T := proc(n, k) option remember; if n=1 then return(1) fi; if kJohannes W. Meijer, Sep 22 2010, revised Jul 17 2013

Mathematica

T[1, ]:= 1; T[n, k_]/;(k
				

Sage

def A033877_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)^k*prec(n, n-k) for k in (0..n-1)]
for n in (1..10): print(A033877_row(n)) # Peter Luschny, Mar 16 2016

SageMath

@CachedFunction
def t(n, k): # t = A033847
    if (k<0 or k>n): return 0
    elif (k==1): return 1
    else: return t(n, k-1) + t(n-1, k-1) + t(n-1, k)
flatten([[t(n,k) for k in range(1,n+1)] for n in range(1, 16)]) # G. C. Greubel, Mar 23 2023

Formula

As an upper right triangle: a(n, k) = a(n, k-1) + a(n-1, k-1) + a(n-1, k) if k >= n >= 0 and a(n, k) = 0 otherwise.

G.f.: Sum T(n, k)*x^n*y^k = (1-x*y-(x^2*y^2-6*x*y+1)^(1/2)) / (x*(2*y+x*y-1+(x^2*y^2-6*x*y+1)^(1/2))). - Vladeta Jovovic, Feb 16 2003

Another version of A000007 DELTA [0, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...] = 1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 6, 16, 22, 0, 1, ..., where DELTA is Deléham's operator defined in A084938.

Sum_{n=1..floor((k+1)/2)} T(n+p-1, k-n+p) = A026003(2*p+k-3) - A008288(2*p+k-3, p-2), p >= 2, k >= 1. - Johannes W. Meijer, Sep 28 2013

From G. C. Greubel, Mar 23 2023: (Start)

(t(n, k) as a lower triangle)

t(n, k) = t(n, k-1) + t(n-1, k-1) + t(n-1, k) with t(n, 1) = 1.

t(n, n) = A006318(n-1).

t(2*n-1, n) = A330801(n-1).

t(2*n-2, n) = A103885(n-1), n > 1.

Sum_{k=1..n-1} t(n, k) = A238112(n), n > 1.

Sum_{k=1..n} t(n, k) = A001003(n).

Sum_{k=1..n-1} (-1)^(k-1)*t(n, k) = (-1)^n*A001003(n-1), n > 1.

Sum_{k=1..n} (-1)^(k-1)*t(n, k) = A080243(n-1).

Sum_{k=1..floor((n+1)/2)} t(n-k+1, k) = A026003(n-1). (End)

Extensions

More terms from David W. Wilson