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A033877 Triangular array 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). 20
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, 1, 20, 198 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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)

LINKS

T. D. Noe, Rows k=1..50 of triangle, flattened

H. Bottomley, Illustration of initial terms

Kevin Brown, Hipparchus on Compound Statements, 1994-2010. - Johannes W. Meijer, Sep 22 2010

G. Kreweras, Sur les hiérarchies de segments, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973.

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.

R. A. Sulanke, Objects counted by the central Delannoy numbers, J. Integer Seq. 6 (2003), Article 03.1.5, 19 pp.

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

EXAMPLE

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

MAPLE

T := proc(n, k) option remember; if n=1 then return(1) fi; if k<n then return(0) fi; T(n, k-1)+T(n-1, k-1)+T(n-1, k) end: seq(seq(T(n, k), n = 1..k), k=1..10); # Johannes W. Meijer, Sep 22 2010, revised Jul 17 2013

MATHEMATICA

T[ 1, _ ] := 1; T[ n_, k_ ]/; (k<n) := 0; T[ n_, k_ ] := T[ n, k ]=T[ n, k-1 ]+T[ n-1, k-1 ]+T[ n-1, k ];

PROG

(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

(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

CROSSREFS

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

Cf. A008288, A006318, A006319, A006320, A006321, A001003 (row sums), A000007, A084938.

Cf. 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).

Sequence in context: A199704 A062344 A208759 * A059369 A199530 A208765

Adjacent sequences:  A033874 A033875 A033876 * A033878 A033879 A033880

KEYWORD

nonn,tabl,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from David W. Wilson

STATUS

approved

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Last modified December 4 23:10 EST 2016. Contains 278755 sequences.