0,2

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

G.f.=[1+z-sqrt(1-6*z+z^2)]^4/(256z^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.

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} ]

Cf. A001003.

Right-hand column 4 of triangle A011117.

Fourth column of convolution triangle A011117.

Sequence in context: A014348 A126020 A086405 this_sequence A007859 A085923 A110166

Adjacent sequences: A010846 A010847 A010848 this_sequence A010850 A010851 A010852

nonn

Robert Sulanke (sulanke(AT)diamond.idbsu.edu)

More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 27 2003