%I A088617
%S A088617 1,1,1,1,3,2,1,6,10,5,1,10,30,35,14,1,15,70,140,126,42,1,21,140,420,630,
%T A088617 462,132,1,28,252,1050,2310,2772,1716,429,1,36,420,2310,6930,12012,
%U A088617 12012,6435,1430,1,45,660,4620,18018,42042,60060,51480,24310,4862,1,55
%N A088617 Triangle T(n,k) (n>=0, k=0..n) read by rows: T(n,k) = C(n+k,n)*C(n,k)/
               (k+1).
%C A088617 Row sums: A006318 (Schroeder numbers). Essentially same as triangle A060693 
               transposed.
%C A088617 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
%C A088617 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
%C A088617 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
%C A088617 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]
%D A088617 Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal 
               Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%D A088617 Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 449.
%D A088617 M. Klazar, On numbers of Davenport-Schinzel sequences, Discr. Math., 
               185 (1998), 77-87.
%D A088617 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]
%H A088617 Laradji, A. and Umar, A., <a href="http://www.cs.uwaterloo.ca/journals/
               JIS/">Combinatorial Results for Semigroups of Order-Decreasing Partial 
               Transformations </a>, Journal of Integer Sequences, Vol. 7 (2004), 
               Article 04.3.8. [From A. Umar (aumarh(AT)squ.edu.om), Oct 02 2008]
%F A088617 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.
%F A088617 T(n, k) = A085478(n, k)*A000108(k); A000108 = Catalan numbers. - DELEHAM 
               Philippe (kolotoko(AT)wanadoo.fr), Dec 05 2003
%F A088617 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
%F A088617 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
%e A088617 Triangle begins:
%e A088617 1
%e A088617 1 1
%e A088617 1 3 2
%e A088617 1 6 10 5
%e A088617 1 10 30 35 14
%e A088617 1 15 70 140 126 42
%o A088617 (PARI) {T(n, k)= if(k+1, binomial(n+k, n)*binomial(n, k)/(k+1))}
%Y A088617 Diagonals give A000217, A034827, A000910, A088625, A088626, A000108, 
               A001700, A002457, A002802, A002803.
%Y A088617 Cf. A084938 A006318 A060693 A085478 A000108.
%Y A088617 Sequence in context: A114586 A052174 A111049 this_sequence A144250 A156367 
               A008276
%Y A088617 Adjacent sequences: A088614 A088615 A088616 this_sequence A088618 A088619 
               A088620
%K A088617 nonn,tabl
%O A088617 0,5
%A A088617 N. J. A. Sloane (njas(AT)research.att.com), Nov 23 2003
%E A088617 More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Nov 23 
               2003