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Search: id:A011117
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| A011117 |
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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. |
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15
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| 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, 550, 903, 1, 7, 33, 125, 403, 1119, 2591, 4279, 1, 8, 42, 176, 630, 1976, 5424, 12536, 20793, 1, 9, 52, 238, 930, 3206, 9860, 26832, 61921, 103049, 1, 10, 63
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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When seen as polynomials with descending coefficients: evaluations are A006318 (x=1), A001003 (x=2).
Triangular array in A104219 transposed. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 16 2005
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
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LINKS
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E. Barcucci, E. Pergola, R. Pinzani and S. Rinaldi, ECO method and hill-free generalized Motzkin paths
E. Pergola and R. A. Sulanke, Schroeder Triangles, Paths and Parallelogram Polyominoes, J. Integer Sequences, 1 (1998), #98.1.7.
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FORMULA
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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).
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.
G.f.: 2/[1+uv-2v+sqrt(1-6uv+u^2v^2)]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 25 2003
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)
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]
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MATHEMATICA
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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} ]
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CROSSREFS
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Cf. A084938.
Right-hand columns show convolutions of little Schroeder numbers with themselves: A001003, A010683, A010736, A010849.
Sequence in context: A071943 A062869 A102473 this_sequence A069269 A100324 A121424
Adjacent sequences: A011114 A011115 A011116 this_sequence A011118 A011119 A011120
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KEYWORD
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nonn,tabl
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AUTHOR
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Robert Sulanke (sulanke(AT)diamond.idbsu.edu)
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