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A010849
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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-3,n).
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2
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1, 4, 18, 84, 403, 1976, 9860, 49912, 255701, 1323292, 6907830, 36331500, 192339687, 1024140336, 5481165832, 29469454640, 159094662121, 862087135988, 4687164401114, 25562520325828, 139803777476859, 766578879858024
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Number of dissections of a convex polygon with n+5 sides that have a pentagon over a fixed side (the base) of the polygon. Example: a(1)=4 because the only dissections of the convex hexagon ABCDEF (AB being the base), that have a pentagon over AB are the dissections made by the diagonals FD, EC, AE 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 3 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))^4/(256*z^4). 4-fold convolution of A001003 with itself. Convolution of A010683 with itself. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 27 2003
a(n) = (4/n)*sum(binomial(n, k)*binomial(n+k+3, k-1), k=1..n) = 4*hypergeom([1-n, n+5], [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 4 of triangle A011117.
Fourth column of convolution triangle A011117.
Sequence in context: A126020 A086405 A151251 * A007859 A185132 A085923
Adjacent sequences: A010846 A010847 A010848 * A010850 A010851 A010852
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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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