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A010736
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Let S(x,y) = number of lattice paths from (0,0) to (x,y) that use the step set { (0,1), (1,0), (2,0), (3,0), ....} and never pass below y = x. Sequence gives S(n-2,n).
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1
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1, 3, 12, 52, 237, 1119, 5424, 26832, 134913, 687443, 3541932, 18421524, 96585597, 509960223, 2709067968, 14469453632, 77655751329, 418567792899, 2264867271852, 12298297439892, 66993811842477, 366009125766463
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Threefold convolution of A001003 with itself. Number of dissections of a convex polygon with n+4 sides that have a quadrilateral over a fixed side (the base) of the polygon. Example: a(1)=3 because the only dissections of the convex pentagon ABCDE (AB being the base), that have a quadrilateral over AB are the dissections made by the diagonals EC, AD and BD, respectively. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 27 2003
a(n-1) = number of royal paths (A006318) from (0,0) to (n,n) with exactly 2 diagonal steps on the line y=x. - David Callan (callan(AT)stat.wisc.edu), Jul 15 2004
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FORMULA
| G.f.=[1+z-sqrt(1-6*z+z^2)]^3/(64z^3). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 27 2003
a(n) = (3/n)*sum(binomial(n, k)*binomial(n+k+2, k-1), k=1..n) = 3*hypergeom([1-n, n+4], [2], -1), n>=1, a(0)=1.
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MATHEMATICA
| 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
| Cf. A001003.
Right-hand column 3 of triangle A011117.
Third column of convolution triangle A011117.
Sequence in context: A007856 A151195 A151196 * A151197 A007198 A000256
Adjacent sequences: A010733 A010734 A010735 * A010737 A010738 A010739
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KEYWORD
| nonn
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AUTHOR
| Robert Sulanke (sulanke(AT)diamond.idbsu.edu)
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EXTENSIONS
| More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 27 2003
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