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%I #24 Feb 17 2014 23:44:55
%S 1,4,18,84,403,1976,9860,49912,255701,1323292,6907830,36331500,
%T 192339687,1024140336,5481165832,29469454640,159094662121,
%U 862087135988,4687164401114,25562520325828,139803777476859,766578879858024
%N 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).
%C 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_, Dec 27 2003
%C 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_, Jul 15 2004
%H Vincenzo Librandi, <a href="/A010849/b010849.txt">Table of n, a(n) for n = 0..200</a>
%F 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_, Dec 27 2003
%F 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 Recurrence: n*(n+4)*a(n) = (7*n^2+16*n-3)*a(n-1) - (7*n^2-2*n-12)*a(n-2) + (n-3)*(n+1)*a(n-3). - _Vaclav Kotesovec_, Oct 07 2012
%F a(n) ~ sqrt(1632+1154*sqrt(2))*(3+2*sqrt(2))^n/(4*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 07 2012
%F Recurrence (an alternative): (n+4)*a(n) = (8-n)*a(n-8) + 4*(2*n-13)*a(n-7) + 12*(5-n)*a(n-6) + 4*(7-2*n)*a(n-5) + 26*(n-2)*a(n-4) + 4*(1-2*n)*a(n-3) - 12*(n+1)*a(n-2) + 4*(2*n+5)*a(n-1), n>=8. - _Fung Lam_, Feb 18 2014
%t 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} ] (* End *)
%t CoefficientList[Series[(1 + x - Sqrt[1 - 6 x + x^2])^4 / (256 x^4), {x, 0, 30}], x] (* _Vincenzo Librandi_, May 03 2013 *)
%o (PARI) x='x+O('x^66); Vec((1+x-sqrt(1-6*x+x^2))^4/(256*x^4)) \\ _Joerg Arndt_, May 04 2013
%Y Cf. A001003.
%Y Right-hand column 4 of triangle A011117.
%Y Fourth column of convolution triangle A011117.
%K nonn
%O 0,2
%A Robert Sulanke (sulanke(AT)diamond.idbsu.edu)
%E More terms from _Emeric Deutsch_, Dec 27 2003
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