%I A011117
%S A011117 1,1,1,1,2,3,1,3,7,11,1,4,12,28,45,1,5,18,52,121,197,1,6,25,84,237,
%T A011117 550,903,1,7,33,125,403,1119,2591,4279,1,8,42,176,630,1976,5424,
%U A011117 12536,20793,1,9,52,238,930,3206,9860,26832,61921,103049,1,10,63
%N A011117 Triangle of numbers S(x,y) = number of lattice paths from (0,0) to (x,
               y) that use step set { (0,1), (1,0), (2,0), (3,0), ....} and never 
               pass below y = x.
%C A011117 When seen as polynomials with descending coefficients: evaluations are 
               A006318 (x=1), A001003 (x=2).
%C A011117 Triangular array in A104219 transposed. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Mar 16 2005
%C A011117 Triangle T(n,k), 0<=k<=n, defined by : T(0,0) = 1, T(n,k) = T(n-1,k) 
               + Sum_{j, 0<=j<=k-1} 2^j*T(n-1,k-1-j) . - Philippe DELEHAM(kolotoko(AT)wanadoo.fr), 
               Oct 10 2005
%H A011117 E. Barcucci, E. Pergola, R. Pinzani and S. Rinaldi, <a href="http://www.mat.univie.ac.at/
               ~slc/opapers/s46rinaldi.html">ECO method and hill-free generalized 
               Motzkin paths</a>
%H A011117 E. Pergola and R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/
               JIS/index.html">Schroeder Triangles, Paths and Parallelogram Polyominoes</
               a>, J. Integer Sequences, 1 (1998), #98.1.7.
%F A011117 S(m, n)=[(n-m+1)/(n+1)]sum(2^(m-i-1)binomial(n+1, i+1)binomial(m-1, i), 
               i=0..m-1).
%F A011117 Another version of triangle [1, 0, 0, 0, 0, 0, ...] DELTA [0, 1, 2, 1, 
               2, 1, 2, 1, 2, 1, ...] = 1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 11, 
               0, 1, 4, 12, 28, 45, 0, 1, ..., where DELTA is Deleham's operator 
               defined in A084938.
%F A011117 G.f.: 2/[1+uv-2v+sqrt(1-6uv+u^2v^2)]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Dec 25 2003
%F A011117 Sum_{k = 0..n} T(n, k) = A006318(n), large Schroeder numbers. - DELEHAM 
               Philippe (kolotoko(AT)wanadoo.fr), Jul 10 2004. (This is because 
               T(n, k) = number of royal paths (A006318) of length n with exactly 
               n-k Northeast steps lying on the line y=x. - David Callan (callan(AT)stat.wisc.edu), 
               Aug 02 2004)
%F A011117 S(n,m) = ((n-m+1)/m)*sum(binomial(m,k)*binomial(n+k,k-1),k=1..m), n>=m>
               1; S(n,0)=1; S(n,m)=0, n<m. See the corresponding formula for A104219. 
               [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Mar 16 2009]
%t A011117 f[ x_, y_ ] := f[ x, y ] = Module[ {return}, If[ x == 0, return = 1, 
               If[ y == x-1, return = 0, return = f[ x, y-1 ] + Sum[ f[ k, y ], 
               {k, 0, x-1} ] ] ]; return ]; Do[ Print[ Table[ f[ k, j ], {k, 0, 
               j} ] ], {j, 10, 0, -1} ]
%Y A011117 Cf. A084938.
%Y A011117 Right-hand columns show convolutions of little Schroeder numbers with 
               themselves: A001003, A010683, A010736, A010849.
%Y A011117 Sequence in context: A071943 A062869 A102473 this_sequence A069269 A100324 
               A121424
%Y A011117 Adjacent sequences: A011114 A011115 A011116 this_sequence A011118 A011119 
               A011120
%K A011117 nonn,tabl
%O A011117 0,5
%A A011117 Robert Sulanke (sulanke(AT)diamond.idbsu.edu)