0,5

Row sums: A006318 (Schroeder numbers). Essentially same as triangle A060693 transposed.

T(n,k) is number of Schroeder paths (i.e. consisting of steps U=(1,1), D=(1,-1),H=(2,0) and never going below the x-axis) from (0,0) to (2n,0), having k U's. E.g. T(2,1)=3 because we have UHD, UDH and HUD. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 06 2003

Little Schroeder numbers A001003 have a(n)=sum{k=0..n, A088617(n,k)*(-1)^(n-k)*2^k} - Paul Barry (pbarry(AT)wit.ie), May 24 2005

Conjecture: The expected number of Us in a Schroeder n-path is asymptotically Sqrt[1/2]*n for large n. - David Callan (callan(AT)stat.wisc.edu), Jul 25 2008

T(n, k) is also the number of order-preserving and order-decreasing partial transformations (of an n-chain) of width k (width(alpha) = |Dom(alpha)|). [From A. Umar (aumarh(AT)squ.edu.om), Oct 02 2008]

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 449.

M. Klazar, On numbers of Davenport-Schinzel sequences, Discr. Math., 185 (1998), 77-87.

Laradji, A. and Umar, A. Combinatorial results for semigroups of order-decreasing partial transformations J. Integer Seq. 7 (2004), 04.3.8, 14 [From A. Umar (aumarh(AT)squ.edu.om), Oct 02 2008]

Laradji, A. and Umar, A., Combinatorial Results for Semigroups of Order-Decreasing Partial Transformations , Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.8. [From A. Umar (aumarh(AT)squ.edu.om), Oct 02 2008]

Triangle T(n, k) read by rows; given by [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] DELTA [[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is Deleham's operator defined in A084938.

T(n, k) = A085478(n, k)*A000108(k); A000108 = Catalan numbers. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 05 2003

Sum_{k, 0<=k<=n} T(n, k)*x^k*(1-x)^(n-k) = A000108(n), A001003(n), A007564(n), A059231(n), A078009(n), A078018(n), A081178(n), A082147(n), A082181(n), A082148(n), A082173(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 18 2005

Sum_{k, 0<=k<=n}T(n,k)*x^k = (-1)^n*A107841(n), A080243(n), A000007(n), A000012(n), A006318(n), A103210(n), A103211(n), A133305(n), A133306(n), A133307(n), A133308(n), A133309(n) for x = -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 18 2007

Triangle begins:

1

1 1

1 3 2

1 6 10 5

1 10 30 35 14

1 15 70 140 126 42

(PARI) {T(n, k)= if(k+1, binomial(n+k, n)*binomial(n, k)/(k+1))}

Diagonals give A000217, A034827, A000910, A088625, A088626, A000108, A001700, A002457, A002802, A002803.

Cf. A084938 A006318 A060693 A085478 A000108.

Sequence in context: A114586 A052174 A111049 this_sequence A144250 A156367 A008276

Adjacent sequences: A088614 A088615 A088616 this_sequence A088618 A088619 A088620

nonn,tabl

N. J. A. Sloane (njas(AT)research.att.com), Nov 23 2003

More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Nov 23 2003